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Discrete Morse Theory on the Loop Space of S2

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Discrete Morse Theory on the Loop Space of S2
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Johnson, Lacey A
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Doctorate ( Ph.D.)
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University of Florida
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Mathematics
Committee Chair:
Knudson,Kevin P
Committee Co-Chair:
Bubenik,Peter
Committee Members:
Keesling,James E
Dranishnikov,Alexander Nikolae
Raup-Krieger,Janice
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5/3/2019

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topology
Mathematics -- Dissertations, Academic -- UF
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This paper aims to explore discrete Morse theory in the context of loop spaces. Given a smooth manifold M, its loop space is the set of closed loops in M based at a fixed point x. This is an infinite-dimensional object, but its topology can be understood using classical smooth Morse theory as demonstrated by Milnor. We focus here on the loop space of the 2-sphere, which has the homotopy type of a CW complex with one cell in each dimension, by describing a discrete vector field with desirable properties. ( en )
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Thesis (Ph.D.)--University of Florida, 2019.
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Adviser: Knudson,Kevin P.
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Co-adviser: Bubenik,Peter.
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by Lacey A Johnson.

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DISCRETEMORSETHEORYONTHELOOPSPACEOF S 2 By LACEYJOHNSON ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2019

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c 2019LaceyJohnson

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ACKNOWLEDGMENTS IamdeeplyindebtedtomyadvisorDr.KevinKnudson.Hehasbeenanextraordinary resourceandhasbeenmorethanwillingtooercounselatamoment'snotice.Additionally, ImustthankDr.JamesE.Keesling,Dr.PeterBubenik,Dr.AlexanderDranishnikov,and Dr.JaniceKrieger.Onecouldnotwishforabetterresearchgroupandithasbeenagreat experiencetocollaboratewiththeseindividuals. Iwouldalsoliketoextendmydeepestgratitudetomyparentsfortheirconstantsupport. Thisdissertationwouldnothavebeenpossiblewithouttheirencouragementtopursuemy interestsandinstillinginmeastrongworkethicandpositiveattitude. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS...................................3 LISTOFFIGURES.....................................5 ABSTRACT.........................................6 CHAPTER 1INTRODUCTION...................................7 2LOOPSPACESANDMORSETHEORY.......................8 3SIMPLICIALSETSANDTHEJAMESCONSTRUCTION..............12 4DISCRETEMORSETHEORY............................19 5DISCRETEMORSETHEORYONSIMPLICIALSETS................22 6 S 2 :THEMAXPAIRING...............................27 7RESULTSOFTHEMAXPAIRING..........................36 8SUMMARY......................................40 9FURTHERRESEARCH................................43 APPENDIX ASAMPLEOFCODE..................................44 BRESULTSOFMAXPAIRING.............................51 LISTOFREFERENCES...................................62 BIOGRAPHICALSKETCH.................................63 4

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LISTOFFIGURES Figure page 3-1Elementary2-simplex [0,1,2] with0-simplices 0 , 1 ,and 2 ..............13 3-2Therealizationofthesimplicialsetofthesphere S 2 consistofonlytwonondegenerate simplices,oneindimension0andtheotherindimension2.Thepicturerepresents theimageofthenondegeneratesimplexofdimension2intherealization.Theentire boundaryofthe2-simpleiscollapsedtotheunique0-simplex............14 4-1Discretevectoreldonthetriangulationofthetorus.................20 5

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DISCRETEMORSETHEORYONTHELOOPSPACEOF S 2 By LaceyJohnson May2019 Chair:KevinKnudson Major:Mathematics ThispaperaimstoexplorediscreteMorsetheoryinthecontextofloopspaces.Givena smoothmanifold M ,itsloopspace M isthesetofclosedloopsin M basedataxedpoint x .Thisisaninnite-dimensionalobject,butitstopologycanbeunderstoodusingclassical smoothMorsetheoryasdemonstratedbyMilnor1963.Wefocushereon S 2 ,whichhas thehomotopytypeofaCWcomplexwithonecellineachdimension,bydescribingadiscrete vectoreldwithdesirableproperties. 6

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CHAPTER1 INTRODUCTION MorseTheoryisapowerfultoolfortheanalysisofthetopologyofmanifolds.Indeed, thetopologyofthemanifoldisdeterminedbythecriticalpointsofaMorsefunctionandthe gradientowlinesbetweenthem.Forman2002introducedadiscreteversionofthistheory ongeneralcellcomplexes.Givenacellcomplex X ,a discreteMorsefunction isareal-valued maponthesetofcellsof X satisfyingcertaincombinatorialconditions.Givensuchafunction, onecancomputealgebraic-topologicalinvariantsofthecomplexsuchasitsEulercharacteristic andintegralhomology.Moreover,tocalculatetheseinvariantsitsucestoconstructadiscrete gradientvectoreldonthecomplexratherthananactualfunction. ThisworkexploresdiscreteMorsetheoryinthecontextofloopspaces.Givenasmooth manifold M ,itsloopspace M isthesetofclosedloopsin M basedataxedpoint x .This isaninnite-dimensionalobject,butitstopologycanbeunderstoodusingclassicalsmooth MorsetheoryasdemonstratedbyMilnor1963.Forexample,theloopspaceofthe2-sphere hasthehomotopytypeofaCW-complexwithonecellineachdimension.Thispaperprovides ananalogueofthisresultfromthepointofviewofdiscreteMorsetheorybyconstructinga discretegradientvectoreldonasimplicialmodelof S 2 . Thispaperisorganizedasfollows.InSection2,weintroducesomebasicdenitionsand resultsfromMorsetheory.InSection3,weconstructasimplicialsetmodelingtheloopspace ofa S 2 .ThisrequiresustoreviewsimplicialsetsandtheJamesconstruction.InSection4, wereviewbasicconceptsofdiscreteMorsetheory.InSection5,weconstructadiscretevector eldontheloopspaceof S 2 ;Milnor'scalculationservesasaguide.InSection6,wedevelop analgorithmtogenerateapartialmatchingintheHassediagraminthesimplicialmodelfor theloopspaceof S 2 .InSection9,wediscusshowwemightextendthistomoregeneralloop spaces. 7

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CHAPTER2 LOOPSPACESANDMORSETHEORY InthissectionweexploreclassicalresultsofMorsetheorybyrstunderstandingbasic denitionsandresultsregardingloops,loopspaces,geodesics,andtheenergyfunction. Denition2.0.1. Let M beasmoothmanifoldandlet p and q betwonotnecessarily distinctpointsof M .Apiecewisesmoothpathfrom p to q isamap ! :[0,1] ! M suchthat 1.thereexistsasubdivision 0= t 0 < t 1 < ... < t k =1 of [0,1] ,sothateach ! j [ t i )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 , t i ] is dierentiableofclass C 1 ,and 2. ! = p and ! = q . Thesetofallpiecewisesmoothpathsfrom p to q isdenoted M ; p , q or M .Wecall thissetthepathspaceof M from p to q .When ! = ! , thespaceiscalledtheloop space. Theloopspaceisaninnite-dimensionalobject,butitstopologymaystillbeunderstood usingclassicalsmoothMorsetheoryasdemonstratedbyMilnor1963. Giventwopoints,theshortestpathbetweenthemonacurvedspacecanbedened usingtheformulaforthelengthofacurveandminimizingthislengthusingthecalculusof variations. Denition2.0.2. Aparametrizedpath ! : I ! M whereIdenotesanyintervalofreal numbers,iscalledageodesiciftheaccelerationvectoreld d dt d ! dt isidenticallyzero.Thusthe velocityvectoreld d ! dt mustbeparallelalong ! . Theorem2.0.3 Milnor1963,Corollary10.7 . Supposethatapath ! :[0, ` ] ! M , parametrizedbyarc-length,haslengthlessthanorequaltothelengthofanyotherpathfrom ! to ! ` .Then ! isageodesic. Ageodesicisageneralizationofthenotionofastraightline"tocurvedspaces." Geodesicsarelocallytheshortestpathbetweenpointsinthespace. Denition2.0.4. Ageodesic :[ a , b ] ! M willbecalledminimalifitslengthislessthanor equaltothelengthofanyotherpiecewisesmoothpathjoiningitsendpoints. 8

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Asucientlysmallsegmentofageodesicisminimal.Alonggeodesicmaynotbe minimal.Forexample,agreatcircleofasphereistheintersectionofthesphereandaplane thatpassesthroughthecenterofthesphere.Agreatcircleisageodesic.However,ifithas lengthgreaterthan ,thenitisnotminimal.Minimalgeodesicsarenotunique.Forexample, twoantipodalpointsonaspherearejoinedbyaninnitenumberofminimalgeodesics.For anytwoantipodalpointsonthesphere,thereisauniqueminimalgeodesicbetweenthem. Therearealsoaninnitenumberofnon-minimalgeodesicsbetweenthem.Thesepathsvaryin thenumberoftimestheywraparoundthesphereandthedirectionoftravelaroundthecurve. Example2.0.5. Suppose p and q arenon-antipodalpointson S n .Dene p 'and q 'tobe antipodesof p and q ,respectively.Thereareaninnitenumberofgeodesicsbetween p and q . Denotethesegeodesics 0 , 1 , 2 ,... denedasfollows.Let 0 betheminimalgeodesicfrom p to q .Let 1 denotethelonggreatcirclearc pq ' p ' q .Let 2 denotethearc pqp ' q ' pq ,andso on.Notethatthesubscript k denotesthenumberoftimesthat p or p 'occursintheinterior of k . Suppose M isaRiemannianmanifoldandxametric g .The length ofavector v 2 TM p willbedenotedby jj v jj = < v , v > 1 = 2 . Denition2.0.6. Let M beaRiemannianmanifoldwithmetric g .If :[0,1] ! M isa piecewise-smoothpath,denetheenergyof tobe E = Z 1 0 d dt 2 dt . Onewaytounderstandtheenergyfunction E istoimaginearubberbandstretched acrossthesurfaceofaball.Pullingthebandtooneside,therebymakingitlonger,requires moreenergy.Milnorshowedthattheenergyfunctionisminimizedalonggeodesicsin M . Theorem2.0.7 Milnor1963,Corollary12.3 . Thepath ! isacriticalpointfortheenergy function E ifandonlyif ! isageodesic. Atthispoint,weneedtotalkabouttheHessianoftheenergyfunctional.Weonlydene theHessianatcriticalpointsof E ;Theorem2.0.7impliesthatthesearethegeodesics. 9

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Denition2.0.8. Givenvectorelds W 1 , W 2 2 T ,let a : U [0,1] ! M beatwo parametervariationwhere U isaneighborhoodoftheoriginin R 2 suchthat ,0, t = t , u 1 ,0, t = W 1 t , u 2 ,0, t = W 2 t thentheHessian E W 1 , W 2 isdenedasfollows E W 1 , W 2 := 2 E u 1 , u 2 u 1 u 2 ,0 Denition2.0.9. Twopoints p , q 2 M areconjugatealong ifthereexistsanonzeroJacobi eld J along whichvanishesat p and q .Themultiplicityof p and q asaconjugatepairis thedimensionofthevectorspaceofsuchJacobields. Denition2.0.10. TheindexoftheHessian E isthemaximumdimensionofasubspaceon which E isnegativedenite. Theorem2.0.11 Milnor1963,Theorem15.1 . Theindex oftheHessian E ,isequalto thenumberofpoints t with 0 < t < 1 suchthat t isconjugateto along ,where wecountconjugatepointswithmultiplicity.Theindex isalwaysnite. Ingeneral,thecriticalpointsof E correspondtogeodesicloops through x ,withthe numberoftimes passesthrough x beingthe index ofthecriticalpoint.Milnorprovedthe followingtheorem. Theorem2.0.12 FundamentalTheoremofMorseTheory,Milnor1963,Theorem17.3 . Let M beacompleteRiemannianmanifold,andlet p , q 2 M betwopointswhichare notconjugatealonganygeodesic.Then M ; p , q hasthehomotopytypeofacountable CW-complexwhichcontainsonecellofdimension foreachgeodesicfrom p to q ofindex . ConsiderExample2.0.5againsuchthat n =2 .Inthissimplecaseofthesphere S 2 thegeodesicsarethegreatcircles.Fixingabasepoint x ,thegeodesicloopsthrough x are obtainedbytraversingagreatcirclepassingthrough xn -manytimesfor n =0,1,2,... . Theorem2.0.12tellsusthat S 2 hasthehomotopytypeofaCW-complexwithonecellin eachdimension. 10

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Thisexampleandtheoremsthatfollowmotivatethispaper.Recalltheexampleof non-antipodalpoints p and q onasphere S n . Sincethepoints p and p 'areconjugateto p withmultiplicity n )]TJ/F20 11.9552 Tf 12.057 0 Td [(1 ,thentheindex k isequalto k n )]TJ/F20 11.9552 Tf 11.956 0 Td [(1 .Thus,Milnorgaveusthefollowingresult. Theorem2.0.13 Milnor1963,Corollary17.4 . Theloopspace S n hasthehomotopy typeofaCW-complexwithonecelleachinthedimensions 0, n )]TJ/F20 11.9552 Tf 11.956 0 Td [(1 , 2 n )]TJ/F20 11.9552 Tf 11.956 0 Td [(1 , 3 n )]TJ/F20 11.9552 Tf 11.955 0 Td [(1 ,... Thistellsusthattheremustbeatleastonegeodesicin S n withindex 0 ,atleastone withindex n )]TJ/F20 11.9552 Tf 11.704 0 Td [(1 , 2 n )]TJ/F20 11.9552 Tf 11.704 0 Td [(1 , 3 n )]TJ/F20 11.9552 Tf 11.704 0 Td [(1 ,andsoon.Forexample, S 2 hasatleastonegeodesic withindex0,1,2, ... Thedescriptionof S 2 isclearinthesmoothMorsetheorysetting. However,thequestionremains,canthisresultbediscretized?Thatis,canweprovethisresult usingdiscreteMorsetheory?Thispaperaimstoanswerthesequestionsbyunderstandingthis storyfromthepointofviewofdiscreteMorsetheorybeginningwiththeparticularspecialcase of S 2 .First,wemustconstructasimplicialmodeloftheloopspaceof S 2 .Thiswillrequirean understandingofsimplicialsetsandtheJamesconstruction.WewillthenusediscreteMorse theorytondadiscreteanalogueofTheorem2.0.13for S 2 . 11

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CHAPTER3 SIMPLICIALSETSANDTHEJAMESCONSTRUCTION TounderstandTheorem2.0.13fromthepointofviewofdiscreteMorsetheory,werst needasimplicialcomplexmodelingtheloopspace S 2 .Luckily,thismaybedescribedvia the Jamesconstruction .TheJamesconstruction JX isasimplicialsetbuiltfromthepoints ofaspace X James1955.SimplicialsetsareageneralizationofsimplicialcomplexesMay 1993.Thedenitionisasfollows. Denition3.0.1. Asimplicialset X consistsofasequenceofsets X n of n -simplicesforeach n 0 ,alongwithfacemaps d i : X n ! X n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 anddegeneracymaps s i : X n ! X n +1 foreach i with 0 i n suchthatthefollowingcombinatorialconditionsaresatised d i d j = d j )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 d i , i < j , {1 d i s j = s j )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 d i ,, i < j , {2 d j s j = d j +1 s j = id , {3 d i s j = s j d i )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 , i > j +1, {4 s i s j = s j +1 s i , i j {5 Thefacemap d i takesan n -simplexandreturnsits i th n )]TJ/F20 11.9552 Tf 12.683 0 Td [(1 -face.Thedegeneracy map s i takesan n -simplexandoutputsthe i thdegenerate n +1 -simplexwiththe i th vertexrepeated.Roughlyspeaking,degeneratesimplicesdon'thavethecorrect"numberof dimensions.Geometrically,degeneratesimplicesarehidden"Riehl2011.Herewedenea degeneratesimplex. Denition3.0.2. Adegeneratesimplexisa [ v i 0 ,..., v i n ] forwhichthe v i j arenotalldistinct and i k i l if k < l .Asimplex x 2 X n iscallednon-degenerateif x cannotbewrittenas s i y for any y 2 X n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 andany i . Forexample,considerthestandard n -simplex [0,1,2,..., n ] ,thereare n +1 degeneracy maps s 0 ,..., s n ,denedby s i [0,1,..., n ]=[0,..., j , j ,..., n ] .Inotherwords, s i [0,1,..., n ] gives theuniquedegenerate n +1 simplexwithonlythe i thvertexrepeated.Thesimpleexample 12

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1 2 0 Figure3-1.Elementary2-simplex [0,1,2] with0-simplices 0 , 1 ,and 2 . picturedisanelementary2-simplex [0,1,2] .Thedegenerate1-simplicesare [0,0] , [1,1] ,and [2,2] .Thenon-degenerate1-simplicesare: [0,1] , [0,2] ,and [1,2] . Wewillalsoneedtoknowhowtodeneproductsofsimplicialsetsandtheirsimplices. Denition3.0.3. Let X and Y besimplicialsets.Theirproduct X Y isdenedbythe followingconditions. 1. X Y n = X n Y n = f x , y j x 2 X n , y 2 Y n g 2.If x , y 2 X , Y n ,then d i x , y = d i x , d i y . 3.If x , y 2 X , Y n ,then s i x , y = s i x , s i y . Denition3.0.4. Let X beasimplicialset.Giveeachset X n thediscretetopology,andlet j n j bethe n -simplexwithitsstandardtopology.Therealization j X j isgivenby j X j = q 1 n =0 X n j n j = where istheequivalencerelationgeneratedbytherelations x , D i p d i x , p for x 2 X n +1 , p 2j n j andtherelations x , S i p s i x , p for x 2 X n )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 , p 2j n j .Here, D i and S i arethefaceinclusionsandcollapsesinducedonthestandardgeometricsimplices. Each X n isaset.So, X n j n j isadisjointcollectionofsimplices,oneforeachelement of X n .Therstrelation, x , D i p d i x , p for x 2 X n +1 ,identiescommonfaces.The second, x , S i p s i x , p for x 2 X n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 ,suppressesthedegeneratesimplices,sincethey're encodedwithinnondegeneratesimplicesanyway. Letusconstructasimplicialsetmodelling S 2 Friedman2012.Let X bethesimplicial setwhoseonlynon-degeneratesimplicesarethe 0 -simplex [0] 2 X 0 and 2 -simplex y = 13

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Figure3-2.Therealizationofthesimplicialsetofthesphere S 2 consistofonlytwo nondegeneratesimplices,oneindimension0andtheotherindimension2.The picturerepresentstheimageofthenondegeneratesimplexofdimension2inthe realization.Theentireboundaryofthe2-simpleiscollapsedtotheunique 0-simplex. [0,1,2] 2 X 2 .Allothersimplicesaredegenerate.So,in X 1 thereisdegeneratesimplex [0,0] . Wealsohavedegeneratesimplices s 0 y , s 1 y , s 2 y ,andsoon.Asasimplicialset, S 2 canbe viewedasone0-simplexandone 2 -simplex y withallotherfacesbeingdegenerate.Thisyields aninnite-dimensionalobject.Whenpassingtothegeometricrealization,allofthedegenerate simplicesdisappear.Thus,thegeometricrealizationofthisspaceisequivalenttothestandard constructionof S 2 asaCWcomplexbycollapsingtheboundaryofa 2 -celltoapoint. Alldegeneratesimplicesgetcollapseddownintothesimplicesofwhichtheyare degeneracies.So,constructingarealizationdependsonlyuponunderstandingwhathappensto thenon-degeneratesimplices.Sincewehaveasimplicialset,wecanassigntoitaloopgroup. Denition3.0.5. Let X beasimplicialset.Theloopgroup GX isthesimplicialgroupwith n -simplicesthefreegroupdenedby G n X = F X n +1 = F s n X n . Here, F Y denotesthefreegroupontheset Y .Thefaceanddegeneracymapsin GX are inducedbythosein X . Theorem3.0.6 GoerssandJardine1999,Corollary5.11 . Theloopgroup GX isweakly homotopyequivalentto X . 14

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Recall,ourconstructionofasimplicialmodelof S 2 consistsofone0-cell,one2-cell,and degeneraciesofthese.So,elementsof X n for n > 0 areasfollows. X 0 = f [0] g X 1 = f [0,0] g X 2 = f [0,1,2] g Thiscontinuesfor n > 0 andfor n 3 ,elementsof X n alsohavethedegeneraciesofthe non-trivial 2 -simplex.Fromthispointforwardlet y =[0,1,2] .Let'sconstructtherstfew dimensionsoftheloopgroup G S 2 .Since X 0 = f [0] g ,then s 0 X 0 = f [0,0] g .Wehavethe following. GS 2 0 = F X 1 = F s 0 X 0 = F f [0,0] g = F f [0,0] g = f e g Nextwehavethat s 1 X 1 = f [0,0,0] g .Therefore, GS 2 1 = F X 2 = F s 1 X 1 = F f [0,0,0], y g = F f [0,0,0] g = F f y g Continuingthisprocess,weobtain GS 2 2 = F X 3 = F s 3 X 2 = F f [0,0,0,0], s 0 y , s 1 y , s 2 y g = F f [0,0,0,0], s 2 y g = F f s 0 y , s 1 y g GS 2 3 = F s 0 s 0 y , s 0 s 1 y , s 1 s 1 y GS 2 4 = F s 0 s 0 s 0 y , s 0 s 0 s 1 y , s 0 s 1 s 1 y , s 1 s 1 s 1 y 15

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GS 2 5 = F s 0 s 0 s 0 s 0 y , s 0 s 0 s 0 s 1 y , s 0 s 0 s 1 s 1 y , s 0 s 1 s 1 s 1 y , s 1 s 1 s 1 s 1 y Thiscontinuesforall n > 0 .Weneedtomakeafewnotesbeforewecontinue.We writetheuniquegeneratorsinthiswaybyusingthedegeneracyrelationfromDenition3.0.1. Specically,relationtellsusthat s i s j = s j +1 s i for i j .Let'sseeafewexamplesofthisat work.Fordimensionthree,wehavegenerators s 0 s 0 y , s 0 s 1 y , and s 1 s 1 y .Wedon'thave s 1 s 0 y listedasageneratorbecausethedegeneracyrelationtellsus s 1 s 0 y = s 0 s 0 y .Fordimension four,wehavetheuniquegenerators s 0 s 0 s 0 y , s 0 s 0 s 1 y , s 0 s 1 s 1 y , and s 1 s 1 s 1 y .Wedon'thave s 0 s 1 s 0 y asageneratorsince s 0 s 1 s 0 y = s 0 s 0 s 0 y . Similarly, s 1 s 0 s 0 y = s 0 s 0 s 0 y s 1 s 0 s 1 y = s 0 s 0 s 1 y s 1 s 1 s 0 y = s 0 s 0 s 0 y Wechoosetheuniquegeneratorsabovebecausethereisanobviouslinearorderonthe generators,tobediscussedbelow.Weintroducethefollowingnotationfortheindices.To simplifynotation,wewilldenote s i 1 s i 2 s i 3 ... s i k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 y as s i 1 i 2 i 3 ... i k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 y .Forexample,insteadofwriting s 0 s 0 s 1 s 1 y wewillwrite s 0011 y . Since GS 2 isafreesimplicialgroup,theelementsare n -simplicesthatconsistofproducts ofthegeneratorsandtheirinverses.Thereareaninnitenumberofsimplicesinthisspace. Wewouldlikeawaytoreducethenumberofsimplicestomakeourcalculationsmore manageable.WeareabletodothatwiththeJamesconstruction. Denition3.0.7. TheJamesconstruction JX ofatopologicalspace X withbasepoint e is thefreesimplicialmonoid JX = q n 1 X n = where x 1 ,... x k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 , e , x k +1 ,... x n x 1 ,... x k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 , x k +1 ,... x n . 16

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Jamesprovedthefollowingtheorem. Theorem3.0.8 James1955 . JX ishomotopyequivalentto j X j when X isconnected. Milnorexpandedonthiswiththeconstructionof FK . Milnorwentontoprovethefollowing. Theorem3.0.9 James1955 . Thegeometricrealizationof JX ,whichwedenote j JX j ,is weaklyhomotopyequivalentto j X j , where X isthesuspensionof X . Forourpurposeswewilllookatthisintermsofthesphere.TheJamesconstruction JS n isthesimplicialsetwhose k thlevelisthefreesimplicialmonoidgeneratedbythe k -simplices in S n .Thatis, JS n k = F S n k modulotherelation s k 0 v =1 .Therefore,weusethe followingtheorem. Theorem3.0.10 Milnor1972 . Thegeometricrealizationof JS n ,whichwedenote j JS n j , isweaklyhomotopyequivalentto j S n j . Thus,theJamesconstructionprovidesasimplicialmodelfortheloopspace,aslongaswe passtothesuspensionoftheoriginalspace.Weusethefactthatthesuspension S 1 = S 2 toconcludethattheloopspaceof S 2 hasasimplicialmodelbasedontheJamesconstruction appliedto S 1 .Thatis, j JS 1 j' j S 1 j = S 2 Thisisafreemonoidgeneratedbyone1-simplex y .Amonoidisasetwithanassociative binaryoperationhavinganidentity.Recall,wewereconsideringtheloopgroupof S 2 ,which isafreegroupwhichcontainsinverses.Nowthatwearedealingwithafreemonoidweno longerhavetoconsidertheinverses.Thisreducesthenumberofsimpliceswehavetowork withconsiderably.Weonlyhavetoconsiderwordswithpositivepowersofthegeneratorsand wecanignoretheinverses. Oursimplicialmodeloftheloopspaceof S 2 isafreemonidgeneratedbyone1-simplex y .Allofitshigherdimensionalsimplicesarefreemonoidsonthedegeneraciesof y .Products ofsuchelementsarenotdegenerateandsoweshouldbeabletondinterestingtopology. Eventhoughtherearedegeneratesimplicesthatarehidden"geometrically,theirproducts 17

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arenotalwaysdegenerate.Forexample,thefreemonoidcontainsthefollowingnondegenerate simplices f y , y 2 , s 0 y s 1 y , s 1 y s 0 y , s 00 y s 11 y , s 01 y s 11 y ... ... y 3 , s 0 y s 0 y s 1 y , s 0 y s 1 y s 1 y ,... g Nowthatwe'vereducedthenumberofsimplicesforoursimplicialmodeloftheloopspace of S 2 ,wewillusediscreteMorsetheorytouncoveradiscreteanalogueofTheorem2.0.13in thesmoothsetting.Todothis,werstneedtoreviewsomebasicsofdiscreteMorsetheory. 18

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CHAPTER4 DISCRETEMORSETHEORY DiscreteMorseTheoryisanadaptationofMorseTheorythatcanbeappliedtosimplicial complexes. Denition4.0.1. Let X beaniteCW-complexanddenoteby K p thesetof p -cellsof X . Wedenoteby K theunionofthe K p .A discreteMorsefunction on X isafunction f : K ! R satisfyingthefollowingforall 2 K p . If isanirregularfaceof p +1 ,then f > f .Also, # f p +1 > : f f g 1 If v p )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 isanirregularfaceof ,then f v < f .Also, # f v p )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 < : f v f g 1 AdiscreteMorsefunctionon X isafunctionwhich,roughlyspeaking,assignshigher numberstohigherdimensionalsimplices,withatmostoneexception,locally,ateachsimplex. Denition4.0.2. Let f beadiscreteMorsefunctionon X .Acell p isa criticalcellof index p ifthefollowingtwoconditionshold: # f p +1 > : f f g =0 # f v p )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 < : f v f g =0 Denition4.0.3. Let X beasimplicialcomplex.Adiscretevectoreld V on X isacollection ofpairs f p < p +1 g ofcellsin X suchthatnocellisinmorethanonepair.Wehave, f p < p +1 g2 V ifandonlyif f f . Wecanvisualizeadiscretevectoreldbydrawinganarrowfrom to .So, f p < p +1 g2 V ifandonlyifwehaveanarrowwith asitstailand asitshead.Inthegure, weseeanexampleofadiscretevectoreldonatriangulationofthetorusKnudson2015. 19

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Figure4-1.Discretevectoreldonthetriangulationofthetorus. Theconditionthatnocellbelongstomorethanonepairin V impliesthatexactlyoneof thefollowingistrueforeachcell : 1. istheheadofexactlyonearrow 2. isthetailofexactlyonearrow 3. isneithertheheadnortailofanarrow Thelastconditionoccurswhen isnotpairedwithanothercell.Inthiscase,wecall a criticalcellof V .Intheexampleofthetorus,wecanseethatmostcellsarepaired.However, ifwestudythegurewecanseetherearefourcriticalcells:one0-cell,two1-cells,andone 2-cell. Denition4.0.4. Let X bearegularCW-complexandsuppose f isadiscreteMorsefunction on X .Thegradientvectoreldof f , r f ,isthediscretevectorelddenedasfollows.If isacriticalcellfor f ,then isacriticalcellfor r f .If p isnotacriticalcellfor f ,then thereiseithera p +1 -cell > with f f ,inwhichcase f < g2r f ,ora 20

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p )]TJ/F20 11.9552 Tf 12.243 0 Td [(1 -cell < with f f ,inwhichcase f < g2r f .Weuse r f forthe gradientbecausethevectorsdrawnon X pointinthedirectionofdecreaseof f . Wecanreturntothegureofthetorusforanexample.Thevectoreldonthetorusis thegradientofadiscreteMorsefunctiononthetorus.Althoughwehavenotconstructeda discreteMorsefunctionforthisexample,thegradientvectoreldgivesusalltheinformation weneedtoknowformostapplications.Therefore,weneednotconstructafunction.Weonly needtondagradientvectoreld. WenowintroducetheHassediagramanditsuseindiscreteMorsetheory. Denition4.0.5. Let X bearegularCW-complex.TheHassediagramof X isthedirected graphwhoseverticesarethecellsin X andedgesarethedirectededgesfrom to for p < p +1 . Denition4.0.6. Let V beadiscretevectoreldon X .If f p < p +1 g2 V ,thenreverse thearrowfrom to intheHassediagram.WecallthisthemodiedHassediagram. ItturnsoutthatwhenamodiedHassediagramhasnodirectedloopsthisproperty characterizesdiscretegradients. Theorem4.0.7 Forman2002 . Adiscretevectoreld V isthegradienteldofadiscrete Morsefunctionon X ifandonlyifthemodiedHassediagramhasnodirectedloops. Thisisthefoundationfortheworkinthispaper.WewanttoconstructamodiedHasse diagramthathasnodirectedloopstothereforeuncoverthegradienteldandtheinformation itgivesus.IfwecanconstructanacyclicpartialmatchingonthemodiedHassediagram constructedon j JS 1 j ,thesimplicialmodelfortheloopspaceof S 2 ,thenitisguaranteedtobe thegradientofadiscreteMorsefunction. 21

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CHAPTER5 DISCRETEMORSETHEORYONSIMPLICIALSETS Wehaveconstructedasimplicialmodelfortheloopspaceof S 2 andwouldliketonda discreteanalogueofMilnor'sclassicresultinsmoothMorsetheoryusingthismodel. Theorem3.0.9tellsusthat j JS 1 j' j S 1 j = S 2 . Therefore,wenolongerhavetoconsiderinversesbecause j JS 1 j isafreemonoidgeneratedby one1-simplex y anditsdegeneracies.Thisspaceconsistsofproductsofdegeneraciesof y that arenotdegeneratethemselves.Intheprevioussection,wefoundthatitsucestoconstruct anacyclicpartialmatchingonthemodiedHassediagramof JS 1 .Weshouldreallythinkof thisasbookkeepingforwhatisgoingoninthegeometricrealizationofthissimplicalset. Inthissection,wewillconstructtheHassediagram.Theverticesarethe p -simplices ofthespace.If p isafaceof p +1 wedenotethis p < p +1 andthereisadirected edgefrom to .Inthenextsection,weconstructamodiedHassediagramfromourHasse diagraminthissection.Thiswillrequireapartialmatching,whichwewillrefertoastheMax Pairing.Thiswillpairsimplices f p < p +1 g andreverseedgesfrom to .Fromtherewe evaluatethepartialmatchingonthemodiedHassediagramandtrytoprovideananalogueof Milnor'sresult. WebeginwiththeconstructionoftheHassediagram.Intheprevioussectionswebuilt JS 1 asthefreemonoidgeneratedbyone1-simplex y anditsiterateddegeneracies.Itcontains y ,productsof y ,andproductsofthedegeneraciesof y thatarenotdegeneratethemselves. ThesesimplicesarethevertricesoftheHassediagram.Thereareaninnitenumberof simplices,butweorganizethembywordlengthanddimension.Thiswillhelpuslateronwhen weconstructourMaxPairing.Inthegraphbelow,wecanbettervisualizehowtoorganizethe manysimplicesintheHassediagram. 22

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y y 2 s 0 ys 1 y s 1 ys 0 y ... s 00 ys 11 y s 01 ys 11 y ... y 3 s 0 ys 0 ys 1 y s 0 ys 1 ys 0 y s 0 ys 1 ys 1 y ... s 00 ys 00 ys 11 y s 00 ys 01 ys 11 y s 00 ys 11 ys 00 y ... WordLength 1 2 3 Dimension 1 2 3 Firstweestablishanorderonthesimplices.Wewillusewordlength,dimension,and theindicesofthegenerators,andthentakethelexicographicorderonthesimplicesofthe Hassediagram.Werstgroupallthesimplicesofsamewordlengthtogether.Wehavethat wordlength n simplicesarelessthanwordlength n +1 simplicesforall n > 0. Forexample, thisgivesusthat y < y 2 < y 3 < ... Withineachwordlength,wegroupallthesimplices ofthesamedimensiontogether.Thatis,wordlength nk -simplicesarelessthanwordlength n k +1 -simplicesforall n , k > 0 .Finally,forallwordlength nk )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplices,thereisan ordergiventousbytheindicesofthegeneratorsoftheproduct.Eachsimplexisaproductof generators.Forall2-simplicesandhigher,thereareindicesonthegeneratorsthatdictatethe orderofthesimplices.The k generatorsofthe k -simplicesarelinearlyordered: s 00 0 y < s 00 01 y < s 00 011 y < < s 011 1 y < s 11 1 y . Wethenorderthe k -simplicesofwordlength n usingtheinducedlexicographicorder. NowthatwehaveestablishedtheverticesoftheHassediagram,weneedtoincorporate edges.Recall,ourspaceisasimplicialsetwithfacemaps d i : X n ! X n )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 forall 0 i n . Thefacemap d i takesinan n -simplexandreturnsits i th n )]TJ/F20 11.9552 Tf 12.126 0 Td [(1 -face.Wehavethefollowing boundaryformulatodeterminethefacesofeachsimplex. 23

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BoundaryFormula Let k bea k -simplexthatistheproductofgeneratorsandhas k +1 -manyfaces f 1 , f 2 ,... f k +1 .Thesewordsaregeneratedby k -manygenerators, k 1 , k 2 ,... k k . Forexample,all2-simplicesaregeneratedby 1 = s 0 y and 2 = s 1 y . Wecallthesethe2-simplexgenerators.All3-simplicesaregeneratedby 1 = s 00 y , 2 = s 01 y , and 3 = s 11 y . Wecallthesethe3-simplexgenerators.All4-simplicesaregeneratedby 1 = s 000 y , 2 = s 001 y , 3 = s 011 y ,and 4 = s 111 y . Wecallthesethe4-simplexgenerators. Observethateachgenerator k i hasindex i thatcorrespondstotheindex j ofthe degeneracymap s j y where j has k )]TJ/F20 11.9552 Tf 12.587 0 Td [(1 digits.Forexample,thegenerator 2 hasindex2, whichdeterminestheindexofthedegeneratesimplex s 01 y .Thenumberoftimes1appearsin j isdeterminedbytheformula i )]TJ/F20 11.9552 Tf 11.955 0 Td [(1. Therestofthedigitsare0. For 1 -simplicesofanywordlength,thesesimplicesaregeneratedby y .Therefore,we beginourdenitionof k -simplexgeneratorsfor n -simplicesofwordlength m where n , m > 1 . Denition5.0.1. For n -simplicesofwordlength m where n , m > 1 ,wedene k )]TJ/F46 11.9552 Tf 9.298 0 Td [(simplex generators tobethedegeneratemapson y thatgenerateall k )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplices.Wewilldenote thesegeneratorsinthefollowingway k 1 , k 2 , k 3 ,..., k i ,..., k k Eachofthesegeneratorsrepresentadegeneratemapof y , s j y ,wheretheindex j has k )]TJ/F20 11.9552 Tf 12.471 0 Td [(1 digits,thelast i )]TJ/F20 11.9552 Tf 12.084 0 Td [(1 digitsoftheindex j are1's,andtheremaining k )]TJ/F20 11.9552 Tf 12.085 0 Td [(1 )]TJ/F20 11.9552 Tf 12.085 0 Td [( i )]TJ/F20 11.9552 Tf 12.085 0 Td [(1= k )]TJ/F41 11.9552 Tf 12.085 0 Td [(i digitsare0. Eachgeneratorappearinginthesimplex k contributesageneratortotheproductin eachofthefaces.Ifageneratorisinthe n thpositionintheproductof k ,thenitisinthe n thpositionintheproductofeachofthefacesof k . 24

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Theminimumgenerator k 1 ,contributes k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 1 tofaces f 1 , f 2 ,... f k andtheidentityto face f k +1 tothe n thpositionoffaceproduct.Forexample, 1 = s 00 ,contributes 1 = s 0 to faces f 1 , f 2 , and f 3 andtheidentitytoface f 4 tothe n thpositionoffaceproduct. Themaximumgenerator k k ,contributes k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 tofaces f 2 , f 3 ,... f k +1 andtheidentity toface f 1 tothe n positionoffaceproduct.Forexample, 3 = s 11 ,contributes 2 = s 1 to faces f 2 , f 3 ,and f 4 andtheidentitytoface f 1 tothe n thpositionofthefaceproduct. Each k i , 2 i k )]TJ/F20 11.9552 Tf 11.893 0 Td [(1 ,contributes k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 i tofaces f 1 , f 2 ,... f k +1 )]TJ/F42 7.9701 Tf 6.586 0 Td [(i ,and k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 i )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 tofaces f k +1 )]TJ/F42 7.9701 Tf 6.586 0 Td [(i +1 ,... f k +1 .Forexample,for 3 = s 011 contributes 3 = s 11 tofaces f 1 and f 2 ,and 2 = s 01 to f 3 and f 4 . Let'slookattheexamplewhen = s 00 s 01 s 11 = 1 2 3 .Thenthefacemapsarethe following: f 1 = 1 2 = s 0 s 1 f 2 = 1 2 2 = s 0 s 1 s 1 f 3 = 1 1 2 = s 0 s 0 s 1 f 4 = 1 2 = s 0 s 1 Atthispoint,itisimportanttoemphasizethata k +1 -simplexgeneratorwillcontribute atmosttwounique k )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplexgeneratorstoitsfacemaps.Theminimumandmaximum generatorswillonlycontributeoneunique k -simplexgeneratortoitsfacemaps.Wecanuse thisfactinreverse.Ifwehavea k -simplexthatisaproductof k -simplexgeneratorswecan determinewhenthissimplexwillbeafaceofa k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplex.Forexample,suppose k i is intheproductofa k -simplex.Theonly k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplexgeneratorsthatwillcontribute k i to thefacesofthe k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplexare k +1 i and k +1 i +1 .Thiswillbeanimportantfactinproving Theorem6.0.1. TheedgesoftheHassediagramwillbedeterminedbythefacemapsofthesimplices. Forallsimplices k ,therewillbeanedgefrom k toeachofitsnon-degeneratefaces.For example, = s 111 s 000 s 001 hasfaces s 00 s 01 , s 11 s 00 s 01 , s 11 s 00 s 01 , s 11 s 00 s 00 , and s 11 s 00 .So, 25

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thereareedges,orarrowsgoingfrom s 111 s 000 s 001 to s 11 s 00 s 01 , s 111 s 000 s 001 to s 11 s 00 s 00 ,and s 111 s 000 s 001 to s 11 s 00 .Thereisnotanedgefrom s 111 s 000 s 001 to s 00 s 01 becausethelatteris degenerate. Let'srecallourobjective.WewouldliketousediscreteMorsetheorytoproveMilnor's theorem.Asarstattempt,onemaytrytondadiscreteanalogueoftheenergyfunctional E .ThiswouldbeadiscreteMorsefunctionon j JS 1 j .Anobviouscandidateforafunctionhere istouse wordlength ineachmonoid.However,onequicklyseesthatthiscannotpossiblybea discreteMorsefunction.Inthenextsection,wetaketheHassediagramwehaveconstructed andmodifyitwithourMaxPairingalgorithmtogenerateadiscretevectoreld.Theorem 4.0.7tellsusthatthisissucient. 26

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CHAPTER6 S 2 :THEMAXPAIRING InthissectionweconstructtheMaxPairingalgorithmthatgeneratesapartialmatching onoursimplicialmodelfortheloopspaceof S 2 denoted JS 2 .Thehopeforthispairingisthat itrecoverstheclassicalresult.Namely,thisdiscretegradienthasasinglecriticalcellineach dimension n =0,1,2,... .Wewillseethatthereareafewthingsthatneedtobeworkedout beforewereachthisconclusion. Let'sbegindescribingtheMaxPairingalgorithm.TheMaxPairingpairsaword-length n k -simplexwiththemaximumword-length n k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplexofwhichitisafaceandwhich hasnotyetbeenpaired.WhenimplementingtheMaxPairingwestartwithwordlengthone simplices,paireverythinginwordlengthone,thenmoveontowordlength2simplices,and soon.Forwordlength n ,westartwith1-simplices,makepossiblepairswith2-simplices, thenmoveonto2-simplicesandmakepossiblepairswith3-simplices,andcontinueinthis waymovingmoveupthewordlengthtohigherdimensions.Whenpairingsimplicestogether, beginwiththemaximum k -simplexandpairitwiththemaximum k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplexofequal wordlengthofwhichitisaregularface.Ifthe k +1 -simplexhasalreadybeenpaired witha k -simplex,thenndthenextlargest k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplexofwhichitisaregularface.If alreadypaired,continuetothenextlargestsimplexuntilapairingismade.Ifnosimplexcan befound,thenleavethe k -simplexunpaired.Anysimplexleftunpairediscritical.Whena pairhasbeenmade,reversetheedgeintheHassediagramfromthe k +1 -simplextothe k -simplex.Remember,the k -simplexisafaceofthe k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplex.Whenweconstructed theHassediagramintheprevioussection,wemadeanedgefromthe k +1 -simplextoits non-degeneratefaces.Therefore,whenwemakeapairwesimplyreversetheedgeconnecting thetwosimplices.Onceasimplexhasbeenpaired,itcannotbepairedagain. Tobemoreclear,let'sdescribetheMaxPairingingreaterdetail.Thealgorithm beginswithword-lengthone1-simplices.Thereisonlyone1-simplex y anditistheonly non-degenerateword-lengthonesimplex.Sinceitistheonlyword-lengthonesimplex,thereis 27

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nosimplextopairitwith.Itremainsunpaired.Hence, y istheuniquecriticalword-lengthone 1-simplex. Sincetherearenomoreword-lengthonesimplices,theMaxPairingmovesonto word-lengthtwosimplices.Itbeginswithdimension1-simplices.Since y 2 istheonlyword lengthtwo1-simplex,webeginthere.Wewanttondthemaximumwordlengthtwo 2-simplexthathas y 2 asafaceandhasnotbeenpreviouslypaired.Thelargest2-simplex thathas y 2 asaregularfaceis s 1 ys 0 y .Therefore,wepairthesetwosimplicestogether f y 2 < s 1 ys 0 y g andreversetheedgesothatwenowhaveanarrowfrom y 2 to s 1 ys 0 y .Since y 2 istheonlyword-lengthtwo1-simplex,wemoveontoword-lengthtwo2-simplices.Webegin withthemaximum2-simplex, s 1 ys 0 y .Sinceitisalreadypairedfrombelow,wemoveonto thenext2-simplex, s 0 ys 1 y .Wewouldliketopairthiswitha3-simplexinwhichitisaregular face.Whenlookingatthe3-simplicesweseethat,although s 0 ys 1 y isafaceofsimplicesin thedimension3,itisnotaregularface.Infact,therearenoword-lengthtwo3-simpliceswith regularfaces.Therefore, s 0 ys 1 y isleftunpaired.Hence, s 0 ys 1 y isacriticalword-lengthtwo 2-simplex. Sincetherearenomoreword-lengthtwo2-simplicestoconsider,wemoveonto word-lengthtwo3-simplices.Sincenoneofthesesimpliceshaveregularfacesnorarethey regularfacesofword-lengthtwo4-simplices,theseareallleftunpairedandarehencecritical. Thisoccursforallword-lengthtwosimplicesofhigherdimension.Therefore,allhigher dimensionalword-lengthtwosimplicescannotbepairedandareleftcritical. SincewehavedonetheMaxPairingforallword-lengthtwosimplices,wemoveon toword-lengththreesimplices.Webeginwithdimensiononesimplices.Thereisonlyone 1-simplex y 3 .Since y 3 istheonlywordlengththree1-simplex,thenwetrytondthe maximumwordlengththree2-simplexthathas y 3 asaregularfaceandhasnotbeenpaired already.Thiswouldbe, s 1 ys 1 ys 0 y .Therefore,wepairthesetwosimplicesupandreversethe edgebetweenthem.Since y 3 istheonly1-simplex,thenwemoveonto2-simplices.Webegin withthemaximum2-simplexandcontinueindescendingorderthroughthe2-simplicesof 28

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wordlengthtwo.Oncedonewith2-simplices,werepeatthesamestepsfor3-simplices,and soon.Aswecontinuepairinghigherdimensions,wereachapointwherenothingcanpair becausetherearenosimpliceswithregularfaces.Forword-lengththreesimplices,thisbegins tohappenindimension8.Werepeatthisprocessforallsimplicesinallwordlengthsandall dimensions. Theorem6.0.1. Let k bethe k -simplexofword-length k consistingoftheproductofthe generatorsinascendingorder: k = k 1 k 2 ... k k . ThentheMaxPairingleaves k critical. Examplesofthisascendingproductleftcriticalare s 0 ys 1 y , s 00 ys 01 ys 11 y ,and s 000 ys 001 ys 011 ys 111 y . Beforeweprovethistheorem,weneedapairoflemmas. Lemma6.0.2. For k -simplexgenerator k i ,theonly k +1 )]TJ/F46 11.9552 Tf 9.299 0 Td [(simplexgeneratorsthatoutput k i asafaceare k +1 i +1 and k +1 i . Proof. ThedescriptionoftheboundaryformulafromSection5isusedtoprovethis.Recall, for k +1 )]TJ/F17 11.9552 Tf 13.201 0 Td [(simplicesthereare k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(manyfaces.Theboundaryformulatellsusthatfor 1 < i < k +1 ,thefaceof k +1 i generatoris k i forfaces f 1 toface f k )]TJ/F42 7.9701 Tf 6.587 0 Td [(i +2 andchanges to k i )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 forfaces f k )]TJ/F42 7.9701 Tf 6.586 0 Td [(i +32 to f k +2 .For i =1 ,thefaceof k +1 1 generatoris k 1 forfaces f 1 toface f k +1 .For i = k +1 ,thefaceof k +1 k +1 generatoris k k forfaces f 2 toface f k +2 . Therefore, k i onlyappearsinthefacemapsof k +1 i +1 and k +1 i . Lemma6.0.3. If k -simplex k isnon-degeneratethenthemaximum k -simplexgenerator k k occursin atleastonce. Proof. Suppose k -simplex k isnon-degenerateandthatitsproductof k )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplexgenerators doesnotcontainthemaximum k )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplexgenerator k k .Let k = k i 1 k i 2 ... k i n {1 = s j 1 y s j 2 y ... s j n y {2 29

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Theindex j l has k )]TJ/F20 11.9552 Tf 12.878 0 Td [(1 -manydigitswiththelast i l )]TJ/F20 11.9552 Tf 12.878 0 Td [(1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(manydigitsbeing1andthe restbeing0.Sincethemaximum k -simplexgeneratorisnotintheproduct,thenallofthe i 1 , i 2 ,..., i n 6 = k .Therefore,each j 1 , j 2 ,..., j n has0asitsrstdigit.Ifwenotatethedigitsof j l , thenfor 1 l n ,let j l = j l 1 j l 2 ... j l k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 .Continuingtheequalitiesabovewegetthefollowing. k = s j 1 1 j 1 2 ... j 1 k )]TJ/F21 5.9776 Tf 5.757 0 Td [(1 ys j 2 1 j 2 2 ... j 2 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 y ... s j n 1 j n 2 ... j n k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 y {3 = s j 1 1 s j 1 2 ... s j 1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 ys j 2 1 s j 2 2 ... s j 2 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 y ... s j n 1 s j n 2 ... s j n k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 y {4 = s 0 s j 1 2 ... s j 1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 ys 0 s j 2 2 ... s j 2 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 y ... s 0 s j n 2 ... s j n k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 y {5 = s 0 s j 1 2 ... s j 1 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 ys j 2 2 ... s j 2 k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 y ... s j n 2 ... s j n k )]TJ/F21 5.9776 Tf 5.756 0 Td [(1 y {6 Wegetline9byexpandingthenotationwehavebeenusing.Recallfromaprevioussection thatwenotated s i 1 s i 2 ... s i k y as s i 1 i 2 ... i k y .Wegetline11byDenition3.0.3.Inparticular, tellsusthat s i ys i y = s i yy .Thus, k isadegeneratesimplex,whichisacontradiction. Therefore, k isaproductof k )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplexgeneratorsthatcontainsthemaximum k )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplex generator k k atleastonce. WearenowreadytoproveTheorem6.0.1. Proof. Inordertoprove k = k 1 k 2 k 2 ... k k iscritical,wemustshowitcannotpair withoneofitsregularfacesthatisawordlength k k )]TJ/F20 11.9552 Tf 12.177 0 Td [(1 -simplex.Wemustalsoshowthat itcannotpairup,andforthisitsucestoshowthatthereisnowordlength k k +1 -simplex ofwhichitisaregularface. Since k = k 1 k 2 k 2 ... k k can'tbepairedfrombelow,theonlywayitcould becriticalisifthereexistsa k +1 -simplexthathas k asaregularfaceandhasnotbeen pairedwitha k -simplexlargerthan k .Wedenotethese k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplicesas k +1 i . First,wemustconsiderthepossible k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplicesthatwillhave k asaface. Wearegiventhat k istheascendingproductof k )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplexgenerators.Thedescription inSection5tellsusthat k +1 i and k +1 i +1 aretheonlygeneratorsthatwillcontributean k i generatortothefaceofa k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplex.ByLemma6.0.2,weknowthepossible 30

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k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(generatorsthatcouldbeintheproductofthe k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplexwearelookingfor. Forexample,therstgeneratorintheproductof k is k 1 .Theonlygeneratorsthatproduce k 1 asafaceare k +1 1 and k +1 2 .Foreachgeneratorintheproductof k thereareonly two k +1 -simplexgeneratorsthatwillproduceitasaface.Belowwehavetheascending productofthe k )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplexgeneratorsandthepossible k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplexgeneratorsthatcould beinthe k +1 -simplexwearelookingfor. k 1 k 2 k 3 ... k k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k k k +1 1 k +1 2 k +1 2 k +1 3 k +1 3 k +1 4 k +1 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k +1 k ... k +1 k k +1 k +1 ByLemma6.0.3,weknowthatthe k +1 -simplexmustcontainthemax k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplex generator, k +1 k +1 ,initsproduct.Theonlytimethatgeneratorappearsasapossibilityisfor k +1 k .Therefore,thelastgeneratorintheproductmustbe k +1 k +1 .Wealterourdiagramto depictthis. k 1 k 2 k 3 ... k k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k k k +1 1 k +1 2 k +1 2 k +1 3 k +1 3 k +1 4 k +1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k +1 k ... k+1 k+1 Allofthegeneratorsintheproductofthe k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplexmustbeunique.A k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplex generatorcannotberepeated.Ifanygeneratorsrepeat,then k willnotbeafaceofthe k + 1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplex.Thiseliminatesseveralpossibilitiesofproductsthatrepresentthe k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplex wearelookingfor.Supposewelettherst k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplexgeneratorintheproductbe k +1 2 . Thenwewouldhavethefollowingdiagram. k 1 k 2 k 3 ... k k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k k k +1 2 k +1 2 k +1 3 k +1 3 k +1 4 k +1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k +1 k ... k +1 k +1 Sincetherecanbenorepeatinggenerators,thenthesecondgeneratorintheproductmustbe k +1 3 .Wecontinuethisprocessuntilwegetthe k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplex: k +1 2 k +1 3 ... k +1 k k +1 k +1 31

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k 1 k 2 k 3 ... k k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k k k +1 2 k +1 3 k +1 4 k +1 k ... k +1 k +1 Nowsupposewechoose k +1 1 astherstgeneratorintheproduct.Thenwehaveachoice forthesecondgenerator.Wecaneitherpick k +1 2 or k +1 3 .Supposewechoose k +1 3 .Since wecan'trepeatgenerators,thethirdgeneratormustbe k +1 4 .Wecontinuethisprocessuntil wegetthe k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplex: k +1 1 k +1 3 k 4 ... k +1 k k +1 k +1 k 1 k 2 k 3 ... k k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k k k +1 1 k +1 3 k +1 4 k +1 k ... k+1 k+1 Noticeintherstexampletheproductskippedthe k +1 1 andinthesecondexamplethe productskipped k +1 2 .Weknowfor k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplicesthereare k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(manygeneratorsthat couldgeneratethesimplex.Wecallthesethe k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplexgenerators.Since k +1 isaword length k simplex,isaproductof k )]TJ/F17 11.9552 Tf 9.298 0 Td [(many k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplexgenerators,andtheproductcannot repeatgenerators,thenatsomepointtheproduct k +1 willexcludeoneofthe k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplex generators.Wesawthisintherstandsecondexamplesabove.Infact,thiswilloccur k -many times.Therefore,wewillhave k -many k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplicesthathave k asaface.Let k +1 s be thesimplexthatskipsthe k +1 s generatorinitsproduct.Ingeneral,thesesimpliceswillbethe followingproducts. k +1 1 = k +1 2 k +1 3 ... k +1 k k +1 k +1 k +1 2 = k +1 1 k +1 3 k +1 4 ... k +1 k k +1 k +1 k +1 3 = k +1 1 k +1 2 k +1 4 k +1 5 ... k +1 k +1 . . . k +1 k = k +1 1 k +1 2 ... k +1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 k +1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k +1 k +1 Theindexofthegeneratorthatgetsskippedwilldeterminewhen k willappearintheface mapsof k +1 s .Thisisbecauseoftheproductofthegeneratorsareinascendingorderandone 32

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ofthe k +1 -simplexgeneratorgetsskipped.Wealsousethebehavioroftheboundaryformula inthat,for 1 < i < k +1 thefaceofan k +1 i generatorchangefrom k i to k i )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 atthe k )]TJ/F41 11.9552 Tf 10.206 0 Td [(i +2 face.Weknowthat k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simpliceshave k +2 )]TJ/F17 11.9552 Tf 9.299 0 Td [(manyfaces.Let s betheindexofthe generatorthatgetsskipped.The k +2 )]TJ/F41 11.9552 Tf 10.577 0 Td [(s = k )]TJ/F41 11.9552 Tf 10.577 0 Td [(s +2 faceandthe k +2 )]TJ/F41 11.9552 Tf 10.577 0 Td [(s +1= k )]TJ/F41 11.9552 Tf 10.577 0 Td [(s +3 faceof k +1 s willbe k .Wecanseethismoreclearlyifwelookatthefacesof k +1 s usingthe boundaryformulawhichdeterminesthefacesofeachsimplex. k +1 s = k +1 1 k +1 2 k +1 3 ... k +1 s )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 k +1 s )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k +1 s +1 k +1 s +2 ... k +1 k +1 f 1 = k 1 k 2 k 3 ... k s )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 k s )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k s +1 k s +2 ... f 2 = k 1 k 2 k 3 ... k s )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 k s )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k s +1 k s +2 ... k k . . . . . . . . . . . . . . . . . . . . . . . . f k )]TJ/F42 7.9701 Tf 6.586 0 Td [(s = k 1 k 2 k 3 ... k s )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 k s )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k s +1 k s +2 ... k k f k )]TJ/F42 7.9701 Tf 6.586 0 Td [(s +1 = k 1 k 2 k 3 ... k s )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 k s )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k s +1 k s +1 ... k k f k )]TJ/F42 7.9701 Tf 6.586 0 Td [(s +2 = k 1 k 2 k 3 ... k s )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 k s )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k s k s +1 ... k k f k )]TJ/F42 7.9701 Tf 6.586 0 Td [(s +3 = k 1 k 2 k 3 ... k s )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 k s )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k s k s +1 ... k k f k )]TJ/F42 7.9701 Tf 6.586 0 Td [(s +2 = k 1 k 2 k 3 ... k s )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 k s )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k s k s +1 ... k k f k )]TJ/F42 7.9701 Tf 6.586 0 Td [(s +3 = k 1 k 2 k 3 ... k s )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 k s )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k s k s +1 ... k k f k )]TJ/F42 7.9701 Tf 6.586 0 Td [(s +4 = k 1 k 2 k 3 ... k s )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 k s )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 k s k s +1 ... k k . . . . . . . . . . . . . . . . . . . . . . . . Since k accountsfortwofacesof k +1 s ,then k isnotaregularfaceofany k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplex. Therefore, k cannotpairwithanythingofdimension k +1 . Nowwewillshow k cannotbepairedfrombelowwithanyofitsregularfacesthatare word-length k k )]TJ/F20 11.9552 Tf 11.955 0 Td [(1 -simplices.Adirectcalculationshowsthatthefacesof k are 33

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k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 . . . k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 2 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 Notethat k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 isanirregularfaceof k ,occurringastherstandlastfaceof k .The remainingfacesareallregularanditsucestoshowthateachonepairswitha k -simplexhigher intheorderthan k .Firstconsidertheface d 1 k = k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 1 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 k )]TJ/F21 7.9701 Tf 6.587 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 . Thisisthemaximalword-length k faceofthe k -simplex k 1 k 2 k k )]TJ/F21 7.9701 Tf 6.586 0 Td [(2 k k k k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 , andisthereforepairedwithitbytheMaxPairingalgorithm.Similarly,theface d 2 k ispaired with k 1 k k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k k )]TJ/F21 7.9701 Tf 6.587 0 Td [(2 k k . Continuinginthismannerweseethat k doesnotgetpairedfrombelowandhenceisacritical simplex. Theorem6.0.4. Let k bethe k -simplexofwordlength k consistingoftheproductofthe generatorsindescendingorderwiththerstgeneratoromittedandthesecondgenerator repeated: k = k k k k )]TJ/F21 7.9701 Tf 6.586 0 Td [(1 k 3 k 2 k 2 . If k isodd,thentheMaxPairingleaves k critical. Proof. TheproofisentirelysimilartothatofTheorem6.0.1.Onerstshowsthat k isnota regularfaceofanyword-length k k +1 -simplex.Foreven k , k getspairedwithoneofits 34

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facesandformspartofacycleintheMaxPairing.When k isodd,theregularfacesof k are pairedwith k +1 -simpliceshigherintheorder.Detailsareomitted. 35

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CHAPTER7 RESULTSOFTHEMAXPAIRING Inthissection,wehavedisplayedsomeresultsoftheMaxPairing.Wedescribethetotal numberofnondegeneratesimplicesandnumberofcriticalsimplicesforwordlengthsone throughveindimensionsonethrougheight.Amoredetailedlistofthecriticalsimplicescan befoundinSectionB. WordLength1 Dimension TotalNumberof Simplices TotalNumberof CriticalSimplices 1 1 1 WordLength2 Dimension TotalNumberof Simplices TotalNumberof CriticalSimplices 1 1 0 2 2 1 3 4 4 4 6 6 5 8 8 6 10 10 7 12 12 8 14 14 36

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WordLength3 Dimension TotalNumberof Simplices TotalNumberof CriticalSimplices 1 1 0 2 6 0 3 18 2 4 36 8 5 60 20 6 90 38 7 126 97 8 168 168 WordLength4 Dimension TotalNumberof Simplices TotalNumberof CriticalSimplices 1 1 0 2 14 0 3 64 0 4 174 1 5 368 16 6 670 66 7 1104 730 8 1694 1694 37

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WordLength5 Dimension TotalNumberof Simplices TotalNumberof CriticalSimplices 1 1 0 2 30 0 3 210 0 4 780 0 5 2100 2 6 4650 32 7 9030 212 8 15960 10261 WeshouldalsomentionthecyclesthatappearoncewecompletetheMaxPairing.Below weseetherstcyclethatappears.Wewillrefertoitas c 1 .Thiscycleinvolveswordlength four3-and4-simplices.AmoredetailedlookatthecyclescanbefoundinSectionB. s 111 ys 011 ys 001 ys 001 y > s 11 ys 01 ys 01 ys 01 y < s 111 ys 011 ys 011 ys 001 y > s 11 ys 11 ys 11 ys 01 y < s 111 ys 111 ys 011 ys 001 y > s 11 ys 11 ys 01 ys 01 y < s 111 ys 011 ys 001 ys 001 y Graphically,thecyclelookslikethis: s 111 ys 011 ys 001 ys 001 y s 11 ys 01 ys 01 ys 01 y s 111 ys 011 ys 011 ys 001 y s 11 ys 11 ys 11 ys 01 y s 111 ys 111 ys 011 ys 001 y s 11 ys 11 ys 01 ys 01 y Therearesomepatternstoobserveaboutthecycles.First,allthecycleslookliketheone abovewiththebottomcornersimplexbeingthemaximalelementintheorderforthatword 38

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length.Forexample, s 11 ys 11 ys 11 ys 01 y isthemaximalelementintheorderforwordlengthfour 3-simplices. Thisrstcycle c 1 involveswordlengthfour3-and4-simplices.Thenextcycle c 2 involves wordlengthfour4-and5-simplices.Wecanobtain c 2 byplacinga1attheendofeach generator.Forexample,in c 1 therstsimplexlistedinthecycleis s 111 ys 011 ys 001 ys 001 y . Addinga1totheendofallthegeneratorsinthisproductproduces s 1111 ys 0111 ys 0011 ys 0011 y , whichistherstsimplexlistedforcycle c 2 .Doingthisforallthesimplicesincycle c 1 yields cycle c 2 .Thispatterncontinuesuptodimension7.Sowehaveacycleinvolvingwordlength four5-and6-simplicesandacycleinvolvingwordlengthfour6-and7-simplices.Cyclesstop appearingafterdimension7becauseindimension8andhighersimplicescannolongerbe paired.Iftherearenopairings,thennoedgesgetreversedandthusnocyclescanexist. Thethirdobservationtobemadeisawaytoconstructcyclesinhigherwordlengths. Consideragaintherstcycle c 1 .Forallthesimplicesinthiscycle,repeatingtherstgenerator producestherstcycleinvolvingwordlengthve3-and4-simplicesdenoted c 5 .Forexample, in c 1 therstsimplexlistedinthecycleis s 111 ys 011 ys 001 ys 001 y .Repeatingtherstgenerator inthisproductyields s 111 ys 111 ys 011 ys 001 ys 001 y ,whichistherstsimplexlistedforcycle c 5 . Dothisforallthesimplicesincycle c 1 andyouwillhavethecycle c 5 .Thispatterncontinues throughdimension7inwordlengthve.Foreverycycleinwordlengthfourthereisacyclein wordlengthvewiththerstgeneratorrepeatedforallthesimplicesinthecycle. InthenextsectionwesummarizeandinterprettheresultsoftheMaxPairingonthe modiedHassediagramofthesimplicialset j JS 1 j . 39

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CHAPTER8 SUMMARY DiscreteMorseTheorytellsusthatamodiedHassediagramwithnodirectedloops isguararanteedtobethegradientofadiscreteMorsefunction.Weconstructedapartial matchingonourmodiedHassediagramconstructedon j JS 1 j bywayoftheMaxPairing.We wouldlikeittobeacyclic;however,aswesawinthelastsection,therearecycles.Weremedy thisinthefollowingway.Eachcycleinvolvesthree k )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplicesandthree k +1 )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplices for 3 k 6 .Ineachcycleinwordlength ` 4 ,thereisanedgegoingfromthemaximal k )]TJ/F17 11.9552 Tf 9.298 0 Td [(simplex ` k tothe k +1 )]TJ/F17 11.9552 Tf 9.299 0 Td [(simplex ` k +1 .Belowwehavethegraphoftherstcycle c 1 . Herewecanseetheboldededgegoingfromthebottomrightsimplex 4 3 = s 11 ys 11 ys 11 ys 01 y tothetoprightsimplex 4 4 = s 111 ys 111 ys 011 ys 001 y . s 111 ys 011 ys 001 ys 001 y s 11 ys 01 ys 01 ys 01 y s 111 ys 011 ys 011 ys 001 y s 11 ys 11 ys 11 ys 01 y s 11 ys 11 ys 11 ys 01 y s 111 ys 111 ys 011 ys 001 y s 111 ys 111 ys 011 ys 001 y s 11 ys 11 ys 01 ys 01 y Toremedythecycleswesimplyremovetheedgethatpairs ` k with ` k +1 .Thiswillgive usanacyclicparitalmatching.However,removingthisedgefromthecycleinproducestwo morecriticalsimplices ` k and ` k +1 .Hereisadepictionofthecriticalsimpliceswegetafter completingtheMaxPairing. 40

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1 1 1 2 2 2 3 3 3 3 4 4 4 3 4 4 4 4 4 5 4 5 4 6 4 6 4 7 5 5 5 3 5 4 5 4 5 5 5 5 5 6 5 6 5 7 4 5 5 6 7 8 Dimension WordLength Wehavefourdierentclassicationsforcriticalsimplices.FromTheorem6.0.1,we knowthat k ,the k -simplexofwordlength k consistingoftheproductofthegenerators inascendingorderisalwaysleftcriticalbytheMaxPairing.Weseethisdepictedinthe graphabove.Thecriticalsimplices k correspondtothecriticalcellsfromMilnor'sTheorem Theorem2.0.13.Wewouldlikethesetobetheonlysimplicesleftcritical.However,thatis notthecase.AlthoughtheMaxPairingisthemostobviouspairingtomake,itleavesother simplicescritical,aswell.WeknowfromTheorem6.0.4, k ,the k -simplexofwordlength k consistingoftheproductofgeneratorsindescendingorderwiththerstgeneratoromittedand thesecondgeneratorrepeated,isalsoleftcriticalbytheMaxPairingwhen k isodd.Wecan seethisinthegraphabove.Inaddition,theMaxPairingleavesuswithcycles.Toridthese cyclesweremoveoneofthepairings,butthatleavesuswithtwomorecriticalsimplicesfor eachcyclethatwehad.Thesearedenotedinthegraphas ` k and ` k where k isthedimension and ` isthewordlengthofthesimplex.Futhermore,foreachwordlength,afteracertain 41

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dimensiontheMaxPairingcannolongerpairsimplicesbecausethesimpliceshavenoregular faces.Therefore,allofthesesimplicesgetleftcritical,aswell. Attheoutset,wehadhopedthatwewouldndacollapsefromourmodelfor S 2 toa cellcomplexwithonecellineachdimension.Wequicklyrealizedthatthiscouldneveroccur forthefollowingreason.If dim word-length ,then willhavenoregularfacesor cofacesandhenceallsimplicesinthisrangemustbeleftcritical.However,theMaxPairing doesgiveusasmallersimplicialmodeloftheloopspaceof S 2 whichmightbeusefulfor computations. InlightoftheworkandresultsofLewineretal2004,weshouldnotbesurprisedthat ouralgorithmfailstoyieldtheexactresultwewereseeking.Indeed,ndinganoptimaldiscrete MorsefunctionisanNP-completeproblem,evenon2-dimensionalsimplicialcomplexes.Given thesimplicityofouralgorithmitwouldhavebeenremarkableifithadgeneratedanoptimal matching. 42

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CHAPTER9 FURTHERRESEARCH Nowthatwehaveamorecompactmodeloftheloopspaceof S 2 wecouldusethissetup tocomputehomology.Thecodeusedforthisstudycanbeusedtocountthegradientpaths amongthecriticalsimplices;thesenumbersgureintheboundarymapinthecorresponding Morsechaincomplex.Asimpliedversionofthecodeusedforthisprojectcanbefoundin SectionA.Inaddition,wewouldliketoreplicatethisworkforhigherdimensionalspheres. Althoughthecurrentprojectisfocusedon S 2 ,thehigherdimensionalspheresshouldbe relativelyeasytoanalyze.Workcouldalsobedonetoextendtheseideasfurthertomore generalspaces.However,ifweareinterestedinspaceswhicharenotspheres,wewillbeforced intoworkingwiththeloopgroup GX ratherthanthesimplicialmonoidemployedhere.The inclusionofinversesin GX complicatesthematterconsiderably. 43

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APPENDIXA SAMPLEOFCODE TheHassediagramthatwemodiedusingtheMaxPairinghadaninnitenumberof simplicesinit.InordertounderstandthisgraphbetteritwasveryusefultocreateaHasse diagraminSageandcomputetheMaxPairingusingthisprogramminglanguage.Belowisan excerptofthecodeusedinSage.First,itcreatestheHassediagram,thenitdoestheMax Pairing. begin; G=DiGraph simpFaceQuant=[] reversedEdges=[] simplices=[] genwl2=[['s0','s1'], ['s00','s01','s11'], ['s000','s001','s011','s111']] genwl3=[['s0','s0','s1','s1'], ['s00','s00','s01','s01','s11','s11'], ['s000','s000','s001','s001', 's011','s011','s111','s111']] genWLList=[genwl2,genwl3] wordlengths=[2,3] 44

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for m in wordlengths: wordLength=m n=m )]TJ/F17 11.9552 Tf 16.149 0 Td [(2 gen=genWLList[n] arrangements=[] for a in gen: A=Arrangementsa,wordLength arrangements.appendA levelSimp=[] a=0 while a < len arrangements: b=arrangements[a] levelsimp=[] for c in b: count=0 for d in c: if d== max gen[a]: count=count+1 if count > 0: simplices.appendc levelsimp.appendc levelSimp.appendlevelsimp a=a+1 for a in simplices: sourceVec=[] for b in a: if b=='s0': 45

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newVec=['y','y',''] elif b=='s1': newVec=['','y','y'] elif b=='s00': newVec=['s0','s0','s0',''] elif b=='s01': newVec=['s1','s1','s0','s0'] elif b=='s11': newVec=['','s1','s1','s1'] elif b=='s000': newVec= ['s00','s00','s00','s00',''] elif b=='s001': newVec= ['s01','s01','s01','s00','s00'] elif b=='s011': newVec= ['s11','s11','s01','s01','s01'] else : newVec= ['','s11','s11','s11','s11'] sourceVec.appendnewVec mergedVec=[] tempStr='' for i inrange 0, len sourceVec[0]: 46

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for j inrange 0, len sourceVec: tempStr=tempStr+sourceVec[j][i] mergedVec.appendtempStr tempStr='' higherVerticeString='' for i inrange 0, len a: higherVerticeString= higherVerticeString+a[i] for a in mergedVec: simpfacequant=[] G.add edgehigherVerticeString,a simpfacequant.appendhigherVerticeString simpfacequant.appenda simpFaceQuant.appendsimpfacequant mergedSimp=[] for a in levelSimp: mergedlevelsimp=[] for b in a: tempStr='' for i inrange 0, len a[0]: tempStr=tempStr+b[i] mergedlevelsimp.appendtempStr mergedSimp.appendmergedlevelsimp mylist= sorted mergedSimp[0] maxMyList= max mylist 47

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facesOfMax= sorted G.neighbors outmaxMyList maxFacesOfMax= max facesOfMax G.reverse edgemaxMyList,maxFacesOfMax reversededges=[] reversededges.appendmaxFacesOfMax reversededges.appendmaxMyList reversedEdges.appendreversededges stop= len mergedSimp )]TJ/F17 11.9552 Tf 9.022 0 Td [(2 t=0 while t < stop: mylist= sorted mergedSimp[t] mylist.reverse mergelength= len mylist k=0 while k < mergelength: maxLevel=mylist[k] neighborsin=G.neighbors inmaxLevel iflen neighborsin > 0: neighborsin= sorted neighborsin smallersimp=[] for a in neighborsin: iflen a < len maxLevel: smallersimp.appenda iflen smallersimp > 0: k=k+1 48

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else : neighborsin.reverse length= len neighborsin l=0 while l < length: alreadypaired= G.neighbors inneighborsin[l] alreadypaired= sorted alreadypaired alreadypairedbelow=[] for c in alreadypaired: iflen c < len neighborsin[l]: alreadypairedbelow.appendc iflen alreadypairedbelow > 0: l=l+1 else : count=0 for b in simpFaceQuant: if b[0]==neighborsin[l]: if b[1]==maxLevel: count=count+1 if count==1: G.reverse edgeneighborsin[l], maxLevel reversededges=[] reversededges.appendmaxLevel 49

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reversededges.appendneighborsin[l ] reversedEdges.appendreversededges l=length else : l=l+1 k=k+1 else : k=k+1 t=t+1 end; 50

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APPENDIXB RESULTSOFMAXPAIRING Inthissection,wehavedisplayedsomeresultsoftheMaxPairing.InSection7,wesaw thenumberofsimplicesandthenumberofcriticalsimplicesforwordlengthsonethroughve indimensionsonethrougheight.Belowwehavelistedthecriticalsimplicesinmoredetail. Criticalsimplicesofwordlengthone: 1-simplices: Thereisone1-simplexanditiscritical.Hereisthecriticalsimplex: y Criticalsimplicesofwordlengthtwo: 1-simplices: Thereisone1-simplexanditisnotcritical. 2-simplices: Therearetwo2-simplicesandoneiscritical.Hereisthecriticalsimplex: s 0 ys 1 y 3-simplices: Therearefour3-simplicesandfourarecritical.Herearethecriticalsimplices: s 00 ys 11 y , s 01 ys 11 y , s 11 ys 00 y , s 11 ys 01 y 4-simplices: Therearesix4-simplicesandsixarecritical.Herearethecriticalsimplices: s 000 ys 111 y , s 001 ys 111 y , s 011 ys 111 y , s 111 ys 000 y , s 111 ys 001 y , s 111 ys 011 y 51

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5-simplices: Thereareeight5-simplicesandeightarecritical.Herearethecriticalsimplices: s 0000 ys 1111 y , s 0001 ys 1111 y , s 0011 ys 1111 y , s 0111 ys 1111 y , s 1111 ys 0000 y , s 1111 ys 0001 y , s 1111 ys 0011 y , s 1111 ys 0111 y 6-simplices: Thereareten6-simplicesandtenarecritical.Herearethecriticalsimplices: s 00000 ys 11111 y , s 00001 ys 11111 y , s 00011 ys 11111 y , s 00111 ys 11111 y , s 01111 ys 11111 y , s 11111 ys 00000 y , s 11111 ys 00001 y , s 11111 ys 00011 y , s 11111 ys 00111 y , s 11111 ys 01111 y 7-simplices: Thereare127-simplicesand12arecritical. 8-simplices: Thereare148-simplicesand14arecritical. Criticalsimplicesofwordlengththree: 1-simplices: Thereisone1-simplexanditisnotcritical. 2-simplices: Therearesix2-simplicesandzeroarecritical. 3-simplices: Thereare183-simplicesandtwoarecritical.Herearethecriticalsimplices: s 00 ys 01 ys 11 y , s 11 ys 01 ys 01 y 4-simplices: Thereare364-simplicesandeightarecritical.Herearethecriticalsimplices: s 000 ys 001 ys 111 y , s 000 ys 011 ys 111 y , s 000 ys 111 ys 001 y , s 001 ys 011 ys 111 y , s 011 ys 000 ys 111 y , s 011 ys 111 ys 000 y , s 111 ys 000 ys 001 y , s 111 ys 011 ys 011 y 52

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5-simplices: Thereare605-simplicesand20arecritical.Herearethecriticalsimplices: s 0000 ys 0001 ys 1111 y , s 0000 ys 0011 ys 1111 y , s 0000 ys 0111 ys 1111 y , s 0000 ys 1111 ys 0001 y , s 0000 ys 1111 ys 0011 y , s 0001 ys 0011 ys 1111 y , s 0001 ys 0111 ys 1111 y , s 0001 ys 1111 ys 0011 y , s 0011 ys 0000 ys 1111 y , s 0011 ys 0111 ys 1111 y , s 0011 ys 1111 ys 0000 y , s 0111 ys 0000 ys 1111 y , s 0111 ys 0001 ys 1111 y , s 0111 ys 1111 ys 0000 y , s 0111 ys 1111 ys 0001 y , s 1111 ys 0000 ys 0001 y , s 1111 ys 0000 ys 0011 y , s 1111 ys 0001 ys 0011 y , s 1111 ys 0011 ys 0000 y , s 1111 ys 0111 ys 0111 y 6-simplices: Thereare904-simplicesand38arecritical.Herearethecriticalsimplices: s 00000 ys 00001 ys 11111 y , s 00000 ys 00011 ys 11111 y , s 00000 ys 00111 ys 11111 y , s 00000 ys 01111 ys 11111 y , s 00000 ys 11111 ys 00001 y , s 00000 ys 11111 ys 00011 y , s 00000 ys 11111 ys 00111 y , s 00001 ys 00011 ys 11111 y , s 00001 ys 00111 ys 11111 y , s 00001 ys 01111 ys 11111 y , s 00001 ys 11111 ys 00011 y , s 00001 ys 11111 ys 00111 y , s 00011 ys 00000 ys 11111 y , s 00011 ys 00111 ys 11111 y , s 00011 ys 01111 ys 11111 y , s 00011 ys 11111 ys 00000 y , s 00011 ys 11111 ys 00111 y , s 00111 ys 00000 ys 11111 y , s 00111 ys 00001 ys 11111 y , s 00111 ys 01111 ys 11111 y , s 00111 ys 11111 ys 00000 y , s 00111 ys 11111 ys 00001 y , s 01111 ys 00000 ys 11111 y , s 01111 ys 00001 ys 11111 y , s 01111 ys 00011 ys 11111 y , s 01111 ys 11111 ys 00000 y , s 01111 ys 11111 ys 00001 y , s 01111 ys 11111 ys 00011 y , s 11111 ys 00000 ys 00001 y , s 11111 ys 00000 ys 00011 y , s 11111 ys 00000 ys 00111 y , s 11111 ys 00001 ys 00011 y , s 11111 ys 00001 ys 00111 y , s 11111 ys 00011 ys 00000 y , s 11111 ys 00011 ys 00111 y , s 11111 ys 00111 ys 00000 y , s 11111 ys 00111 ys 00001 y , s 11111 ys 01111 ys 01111 y , 53

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7-simplices: Thereare1267-simplicesand97arecritical. 8-simplices: Thereare1688-simplicesand168arecritical. Criticalsimplicesofwordlengthfour: 1-simplices: Thereisone1-simplexanditisnotcritical. 2-simplices: Thereare142-simplicesandzeroarecritical. 3-simplices: Thereare643-simplicesandzeroarecritical. 4-simplices: Thereare1744-simplicesandoneiscritical.Herearethecriticalsimplices: s 000 ys 001 ys 011 ys 111 y 5-simplices: Thereare3685-simplicesand16arecritical.Herearethecriticalsimplices: s 0000 ys 0001 ys 0011 ys 1111 y , s 0000 ys 0001 ys 0111 ys 1111 y , s 0000 ys 0001 ys 1111 ys 0011 y , s 0000 ys 0011 ys 0111 ys 1111 y , s 0000 ys 0111 ys 0001 ys 1111 y , s 0000 ys 0111 ys 1111 ys 0001 y , s 0000 ys 1111 ys 0001 ys 0011 y , s 0001 ys 0011 ys 0111 ys 1111 y , s 0011 ys 0000 ys 0111 ys 1111 y , s 0011 ys 0111 ys 0000 ys 1111 y , s 0011 ys 0111 ys 1111 ys 0000 y , s 0111 ys 0000 ys 0001 ys 1111 y , s 0111 ys 0000 ys 1111 ys 0001 y , s 0111 ys 1111 ys 0000 ys 0001 y , s 1111 ys 0000 ys 0001 ys 0011 y , s 1111 ys 0111 ys s 0011 ys 0001 y , 6-simplices: Thereare6706-simplicesand66arecritical.Herearethecriticalsimplices: s 00000 ys 00001 ys 00011 ys 11111 y , s 00000 ys 00001 ys 00111 ys 11111 y , s 00000 ys 00001 ys 01111 ys 11111 y , s 00000 ys 00001 ys 11111 ys 00011 y , 54

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s 00000 ys 00001 ys 11111 ys 00111 y , s 00000 ys 00011 ys 00111 ys 11111 y , s 00000 ys 00011 ys 01111 ys 11111 y , s 00000 ys 00011 ys 11111 ys 00111 y , s 00000 ys 00111 ys 00001 ys 11111 y , s 00000 ys 00111 ys 01111 ys 11111 y , s 00000 ys 00111 ys 11111 ys 00001 y , s 00000 ys 01111 ys 00001 ys 11111 y , s 00000 ys 01111 ys 00011 ys 11111 y , s 00000 ys 01111 ys 11111 ys 00001 y , s 00000 ys 01111 ys 11111 ys 00011 y , s 00000 ys 11111 ys 00001 ys 00011 y , s 00000 ys 11111 ys 00001 ys 00111 y , s 00000 ys 11111 ys 00011 ys 00111 y , s 00000 ys 11111 ys 00111 ys 00001 y , s 00001 ys 00011 ys 00111 ys 11111 y , s 00001 ys 00011 ys 01111 ys 11111 y , s 00001 ys 00011 ys 11111 ys 00111 y , s 00001 ys 00111 ys 01111 ys 11111 y , s 00001 ys 01111 ys 00011 ys 11111 y , s 00001 ys 01111 ys 11111 ys 00011 y , s 00001 ys 11111 ys 00011 ys 00111 y , s 00011 ys 00000 ys 00111 ys 11111 y , s 00011 ys 00000 ys 01111 ys 11111 y , s 00011 ys 00000 ys 11111 ys 00111 y , s 00011 ys 00111 ys 00000 ys 11111 y , s 00011 ys 00111 ys 01111 ys 11111 y , s 00011 ys 00111 ys 11111 ys 00000 y , s 00011 ys 01111 ys 00000 ys 11111 y , s 00011 ys 01111 ys 11111 ys 00000 y , s 00011 ys 11111 ys 00000 ys 00111 y , s 00011 ys 11111 ys 00111 ys 00000 y , s 00111 ys 00000 ys 00001 ys 11111 y , s 00111 ys 00000 ys 01111 ys 11111 y , s 00111 ys 00000 ys 11111 ys 00001 y , s 00111 ys 00001 ys 01111 ys 11111 y , s 00111 ys 01111 ys 00000 ys 11111 y , s 00111 ys 01111 ys 00001 ys 11111 y , s 00111 ys 01111 ys 11111 ys 00000 y , s 00111 ys 01111 ys 11111 ys 00001 y , s 00111 ys 11111 ys 00000 ys 00001 y , s 01111 ys 00000 ys 00001 ys 11111 y , 55

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s 01111 ys 00000 ys 00011 ys 11111 y , s 01111 ys 00000 ys 11111 ys 00001 y , s 01111 ys 00000 ys 11111 ys 00011 y , s 01111 ys 00001 ys 00011 ys 11111 y , s 01111 ys 00001 ys 11111 ys 00011 y , s 01111 ys 00011 ys 00000 ys 11111 y , s 01111 ys 00011 ys 11111 ys 00000 y , s 01111 ys 11111 ys 00000 ys 00001 y , s 01111 ys 11111 ys 00000 ys 00011 y , s 01111 ys 11111 ys 00001 ys 00011 y , s 01111 ys 11111 ys 00011 ys 00000 y , s 11111 ys 00000 ys 00001 ys 00011 y , s 11111 ys 00000 ys 00001 ys 00111 y , s 11111 ys 00000 ys 00011 ys 00111 y , s 11111 ys 00000 ys 00111 ys 00001 y , s 11111 ys 00001 ys 00011 ys 00111 y , s 11111 ys 00011 ys 00000 ys 00111 y , s 11111 ys 00011 ys 00111 ys 00000 y , s 11111 ys 00111 ys 00000 ys 00001 y , s 11111 ys 01111 ys 00111 ys 00011 y , 7-simplices: Thereare1,1047-simplicesand730arecritical. 8-simplices: Thereare1,6948-simplicesand1,694arecritical. Criticalsimplicesofwordlengthve: 1-simplices: Thereisone1-simplexanditisnotcritical. 2-simplices: Thereare302-simplicesandzeroarecritical. 3-simplices: Thereare2103-simplicesandzeroarecritical. 4-simplices: Thereare7804-simplicesandzeroarecritical. 5-simplices: Thereare2,1005-simplicesandtwoarecritical.Herearethecriticalsimplices: s 0000 ys 0001 ys 0011 ys 0111 ys 1111 y 56

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s 1111 ys 0111 ys 0011 ys 0001 ys 0001 y 6-simplices: Thereare4,6506-simplicesand32arecritical.Herearethecriticalsimplices: s 00000 ys 00001 ys 00011 ys 00111 ys 11111 y , s 00000 ys 00001 ys 00011 ys 01111 ys 11111 y , s 00000 ys 00001 ys 00011 ys 11111 ys 00111 y , s 00000 ys 00001 ys 00111 ys 01111 ys 11111 y , s 00000 ys 00001 ys 01111 ys 00011 ys 11111 y , s 00000 ys 00001 ys 01111 ys 11111 ys 00011 y , s 00000 ys 00001 ys 11111 ys 00011 ys 00111 y , s 00000 ys 00011 ys 00111 ys 01111 ys 11111 y , s 00000 ys 00111 ys 00001 ys 01111 ys 11111 y , s 00000 ys 00111 ys 01111 ys 00001 ys 11111 y , s 00000 ys 00111 ys 01111 ys 11111 ys 00001 y , s 00000 ys 01111 ys 00001 ys 00011 ys 11111 y , s 00000 ys 01111 ys 00001 ys 11111 ys 00011 y , s 00000 ys 01111 ys 11111 ys 00001 ys 00011 y , s 00000 ys 11111 ys 00001 ys 00011 ys 00111 y , s 00001 ys 00011 ys 00111 ys 01111 ys 11111 y , s 00011 ys 00000 ys 00111 ys 01111 ys 11111 y , s 00011 ys 00111 ys 00000 ys 01111 ys 11111 y , s 00011 ys 00111 ys 01111 ys 00000 ys 11111 y , s 00011 ys 00111 ys 01111 ys 11111 ys 00000 y , s 00111 ys 00000 ys 00001 ys 01111 ys 11111 y , s 00111 ys 00000 ys 01111 ys 00001 ys 11111 y , s 00111 ys 00000 ys 01111 ys 11111 ys 00001 y , s 00111 ys 01111 ys 00000 ys 00001 ys 11111 y , s 00111 ys 01111 ys 00000 ys 11111 ys 00001 y , s 00111 ys 01111 ys 11111 ys 00000 ys 00001 y , s 01111 ys 00000 ys 00001 ys 00011 ys 11111 y , s 01111 ys 00000 ys 00001 ys 11111 ys 00011 y , s 01111 ys 00000 ys 11111 ys 00001 ys 00011 y , s 01111 ys 11111 ys 00000 ys 00001 ys 00011 y , s 11111 ys 00000 ys 00001 ys 00011 ys 00111 y , s 11111 ys 01111 ys 00111 ys 00011 ys 00011 y , 7-simplices: Thereare9,0307-simplicesand212arecritical. 8-simplices: Thereare15,9608-simplicesand10,261arecritical. 57

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WehavenowseenwhatcriticalsimplicesareleftoverafterourMaxPairing.Nowlet's takealookatthecyclesthatappearoncewecompleteourMaxPairing.Listedbeloware thecyclesthatappearinthemodiedHasseDiagram.Therearesomepatternstotakenote ofaboutthesecycles.Therstcycle c 1 listedinvolveswordlengthfour3-and4-simplices. Noticethattakingthiscycleandplacinga1attheendofeachgeneratoryieldsthenextcycle c 2 thatappearsinwordlengthfourinvolving4-and5-simplices.Forexample,in c 1 therst simplexlistedinthecycleis s 111 ys 011 ys 001 ys 001 y .Addinga1totheendofallthegenerators inthisproductproduces s 1111 ys 0111 ys 0011 ys 0011 y ,whichistherstsimplexlistedforcycle c 2 . Doingthisforallthesimplicesincycle c 1 yieldsthecycle c 2 .Thispatterncontinuesthrough dimension7.Cyclesstopappearingafterdimension7becauseindimension8andhigher simplicescannolongerbepaired.Iftherearenopairings,thennoedgesgetreversedandthus nocyclescanexist. Also,consideragaintherstcycleinvolvingwordlengthfour3-and4-simplices c 1 . Forallthesimplicesinthiscycle,repeatingtherstgeneratorproducestherstcycle involvingwordlengthve3-and4-simplices c 5 .Forexample,in c 1 therstsimplexlisted inthecycleis s 111 ys 011 ys 001 ys 001 y .Ifyourepeattherstgeneratorinthisproductyouget s 111 ys 111 ys 011 ys 001 ys 001 y ,whichistherstsimplexlistedforcycle c 5 .Dothisforallthe simplicesincycle c 1 andyouwillhavethecycle c 5 .Thispatterncontinuesthroughdimension 7.Foreverycycleinwordlengthfourthereisacycleinwordlengthvewiththerst generatorrepeatedforallthesimplicesinthecycle. BelowarethersteightcyclesfoundintheHasseDiagram.Therstcycle c 1 involves wordlengthfour3-and4-simplices. s 111 ys 011 ys 001 ys 001 y > s 11 ys 01 ys 01 ys 01 y < s 111 ys 011 ys 011 ys 001 y > s 11 ys 11 ys 11 ys 01 y < s 111 ys 111 ys 011 ys 001 y > s 11 ys 11 ys 01 ys 01 y < s 111 ys 011 ys 001 ys 001 y Graphically,thecycle c 1 lookslikethis: 58

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s 111 ys 011 ys 001 ys 001 y s 11 ys 01 ys 01 ys 01 y s 111 ys 011 ys 011 ys 001 y s 11 ys 11 ys 11 ys 01 y s 111 ys 111 ys 011 ys 001 y s 11 ys 11 ys 01 ys 01 y Itshouldbenotedthatgraphicallyallthecycleslooklikethiswiththebottomcorner simplexbeingthemaximalelementintheorderforthatwordlength.Forexample, s 11 ys 11 ys 11 ys 01 y isthemaximalelementintheorderforwordlengthfour3-simplices.Onceacycleappearsin thewordlength,itpersistsineachdimensionasdescribedabove. Thesecondcycle c 2 involveswordlengthfour4-and5-simplices. s 1111 ys 0111 ys 0011 ys 0011 y > s 111 ys 011 ys 011 ys 011 y < s 1111 ys 0111 ys 0111 ys 0011 y > s 111 ys 111 ys 111 ys 011 y < s 1111 ys 1111 ys 0111 ys 0011 y > s 111 ys 111 ys 011 ys 011 y < s 1111 ys 0111 ys 0011 ys 0011 y Thethirdcycle c 3 involveswordlengthfour5-and6-simplices. s 11111 ys 01111 ys 00111 ys 00111 y > s 1111 ys 0111 ys 0111 ys 0111 y < s 11111 ys 01111 ys 01111 ys 00111 y > s 1111 ys 1111 ys 1111 ys 0111 y < s 11111 ys 11111 ys 01111 ys 00111 y > s 1111 ys 1111 ys 0111 ys 0111 y < s 11111 ys 01111 ys 00111 ys 00111 y Thefourthcycle c 4 involveswordlengthfour6-and7-simplices. s 111111 ys 011111 ys 001111 ys 001111 y > s 11111 ys 01111 ys 01111 ys 01111 y < s 111111 ys 011111 ys 011111 ys 001111 y > s 11111 ys 11111 ys 11111 ys 01111 y 59

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< s 111111 ys 111111 ys 011111 ys 001111 y > s 11111 ys 11111 ys 01111 ys 01111 y < s 111111 ys 011111 ys 001111 ys 001111 y Thenextcycle c 5 involveswordlengthve3-and4-simplices. s 111 ys 111 ys 011 ys 001 ys 001 y > s 11 ys 11 ys 01 ys 01 ys 01 y < s 111 ys 111 ys 011 ys 011 ys 001 y > s 11 ys 11 ys 11 ys 11 ys 01 y < s 111 ys 111 ys 111 ys 011 ys 001 y > s 11 ys 11 ys 11 ys 01 ys 01 y < s 111 ys 111 ys 011 ys 001 ys 001 y Thenextcycle c 6 involveswordlengthve4-and5-simplices. s 1111 ys 1111 ys 0111 ys 0011 ys 0011 y > s 111 ys 111 ys 011 ys 011 ys 011 y < s 1111 ys 1111 ys 0111 ys 0111 ys 0011 y > s 111 ys 111 ys 111 ys 111 ys 011 y < s 1111 ys 1111 ys 1111 ys 0111 ys 0011 y > s 111 ys 111 ys 111 ys 011 ys 011 y < s 1111 ys 1111 ys 0111 ys 0011 ys 0011 y Thenextcycle c 7 involveswordlengthve5-and6-simplices. s 11111 ys 11111 ys 01111 ys 00111 ys 00111 y > s 1111 ys 1111 ys 0111 ys 0111 ys 0111 y < s 11111 ys 11111 ys 01111 ys 01111 ys 00111 y > s 1111 ys 1111 ys 1111 ys 1111 ys 0111 y < s 11111 ys 11111 ys 11111 ys 01111 ys 00111 y > s 1111 ys 1111 ys 1111 ys 0111 ys 0111 y < s 11111 ys 11111 ys 01111 ys 00111 ys 00111 y Thenextcycle c 8 involveswordlengthve6-and7-simplices. s 111111 ys 111111 ys 011111 ys 001111 ys 001111 y 60

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> s 11111 ys 11111 ys 01111 ys 01111 ys 01111 y < s 111111 ys 111111 ys 011111 ys 011111 ys 001111 y > s 11111 ys 11111 ys 11111 ys 11111 ys 01111 y < s 111111 ys 111111 ys 111111 ys 011111 ys 001111 y > s 11111 ys 11111 ys 11111 ys 01111 ys 01111 y < s 111111 ys 111111 ys 011111 ys 001111 ys 001111 y 61

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LISTOFREFERENCES FormanRAuser'sguidetodiscretemorsetheory.SeminaireLotharingiende Combinatoire,48,ArticleB48c,35pp FriedmanG2Surveyarticle:anelementaryillustratedintroductiontosimplicialsets. RockyMountainJournalofMathematics,42:353{423 GoerssP,JardineJSimplicialHomotopyTheory,ProgressinMathematics,174thedn. Birkhauser,Chicago,IL JamesIReducedproductspaces.AnnofMath62:170{197 KnudsonKMorsetheory:SmoothandDiscrete.WorldSciencePublishingCo.,New Jersey,USA LewinerT,LopesH,TavaresGApplicationsofforman'sdiscretemorsetheoryto topologyvisualizationandmeshcompression.IEEETransactionsonVisualizationand ComputerGraphics5:499{508 MayJPSimplicialobjectsinalgebraictopology,2ndedn.UniversityofChicagoPress, Chicago,IL MilnorJ63Morsetheory.PrincetonUniversityPress,Princeton,NewJersey MilnorJ72Ontheconstrucion FK .AlgebraicTopology-AStudent'sGuide,JFAdams, ed,LondonMathSociety,LectureNotes4,CambridgeUniversityPress,pp118{136 RiehlEAleisurelyintroductiontosimplicialsets.Unpublishedexpositoryarticle availableonlineatURLhttp://www.math.jhu.edu/ eriehl/ssets.pdf 62

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BIOGRAPHICALSKETCH LaceyJohnsonisoriginallyfromStaord,Virginia.ShegraduatedfromBrookePointHigh Schoolin2010.LaceythenattendedJamesMadisonUniversity.Herinterestinmathematical researchgrewin2009whenshecompletedanundergraduateresearchprojectwithElizabeth ArnoldonK-PotentGrobnerBasesandSudoku.ShecomputedGrobnerbasesforasystem ofpolynomialsthatrepresentedtheconstraintsofasmallerversionofSudoku,knownas Shidoku.ShetesteddierentsystemsofpolynomialsinMapletoimprovecomputationtime. ShepresentedthisresearchtoPresidentAlgerofJamesMadisonUniversity,mathfaculty, andothersummerresearchstudents.Shethenwentontopresenthersummerresearchat theShenandoahUndergraduateMathematicsandStatisticsConferenceatJamesMadison UniversityandwonrstplaceintheResearchPosterCompetition.Herresearcheortsalso wonherJamesMadisonUniversity'sMathematicsResearchAward.ShegraduatedJames MadisonUniversitywithaBachelor'sdegreeinmathematicsandcommunicationstudieswitha concentrationininterpersonalcommunicationinMay2014.LaceythenmovedtoGainesville, FloridatoenrollinaMaster'sprograminmathematicsattheUniversityofFlorida,received herMaster'sdegreeinmathematics,andbeganworkonherdoctoraldegreeattheuniversity. Whileattheuniversity,herinterestintopologyanddiscreteMorsetheorygrew.Sheworked withKevinKnudsonandsubmittedapaperonthemin-maxtheoryforcellcomplexes.She thenwentontocontinuestudyingdiscreteMorsetheoryinthecontextofloopspaces.She presentedherresearchattheUF/FSUTopologyandGeometryMeetinghostedatFloridaState Universityin2019.ShewasawardedtheEleanorEwingEhrlichAwardin2017andtheNeil WhiteExcellenceinTeachingAwardin2018.LaceyreceivedherPh.D.fromtheUniversity ofFloridainMay2019.Outsideofmathematics,sheenjoyshikingtrails,boating,traveling, spendingtimewithfamily,andhelpingwomenwiththeirhealthandtness. 63