MA5209 Algebraic Topology

Wayne LawtonDepartment of Mathematics

National University of Singapore

S14-04-04, matwml@nus.edu.sg

http://math.nus.edu.sg/~matwml

Lecture 6. ApplicationsDegrees of Maps of Spheres, Antipodal Preserving Maps

Borsuk-Ulam and Lusternik-Schnirelmann Theorems

Ham Sandwich Theorem

Euler-Poincare and Hopf Trace Formulae

Lefschetz Fixed Point and Lefschetz-Hopf Index Formula

Tor Functor and the Universal Coefficient Theorem

Hopf’s Homotopy Theorem

(27, 30 October 2009)

Maps of SpheresDefinition The degree of a map 1,: nSSf nn

such that Zfd )(is

Questions Let

.)(,)())(( ZSHaafdafH nnn

)(sd

Prove that Is )( fd a homotopy invariant?

For each Zm and 1n constructnn SSf : such that .)( mfd

If 1,:, nSSgf nn and )()( gHfH then are f and g necessarily homotopic?

Show that )()()(1, fdgdgfdnSSS ngnfn

.|| nSK Explain the geometric meaning of

||||: KKs m is a simplicial map.

if

} of facesproper { 1 nK so

Triangulation of SpheresDefinition Simplicial complex is the boundary of the convex set spanned by the set

1 nn R

where

is the standard basis for

1v

Figure n = 1

11:, nivvvvert iiin

11: nivi .1nR

1v

2v

2v

Questions Show that

The set of q-simplices in

1||:1

1

1

n

i in

n xRx

nn S

n is}1||||1:{ 11121

niivvv qiii q

How many q-simplices are there?

triangulation nn Sh ||:

Triangulation of SpheresLemma The chain defined by )( nnn Cz

generates

Proof

.)()( ZHZ nnnn

1,...,1,1)1(2121

121

121][

n

nsss

snssnn vvvsssz

where

1

1

1)1(n

kk

kn Tz

Question Show that every chain is a multiple of nz

1,...,1,1)1(121

121

11]ˆ[

n

nksss

snkssnk vvvsssT

.0]ˆ[1...,1,1,...,1

)1(1211 1111

11

nkk

nk

k sssssnkssn

sk vvvssss

Antipodal Maps

Theorem

simplicial map defined by its values on vertices

Definition The antipodal map

Proof

1,...,1,1)1(2121

121

121][)(

n

nsss

inssnns vvvsssz

.)1()( 1 nsd

||||: nns

Then

is the

.1,...,1,)( nivvvs iii

nn SShsh :1 satisfies nSxxxhsh ,)(1

s induces the chain map )()(: nns CC

and

.)1(][)())(( 1

1,...,1,1)1(2121

121

121 nn

ssssnssn zvvvsss

n

n

Antipodal Maps

Corollary A map

homotopic to the antipodal mapLemma A map

Proof Construct the homotopy nn SSF ]1,0[:

nn SSf : with no fixed points is.:1 nn SShsh

degree

].1,0[,,)()1(

)()1(),(

tSxtxxft

txxfttxF n

nn SSf : with no fixed points has

.)1()( 1 nfd

Corollary If n is even and nn SSf : is homotopic

to nS1 then f has a fixed point.

Vector Fields on Spheres

vector field

Theorem There exists a nonvanishing tangent),0)(( nSxxvx

is odd.

nif and only if

nn SSF ]1,0[:

Proof Let.||,)(||/)()( nSxxvxvxg

be a nonvanishing tangent

nS1is a homotopy from to

.odd is )1()( 1 ngd n

Since,g

defined by

1: nn RSv

1: nn RSv

vector field and let

)()1(

)()1(),(

xtvxt

xtvxttxF

Since

.1)1()( nSdgd

g has no fixed points (why) For odd n the converse is proved by constructing

.),,,...,,(),,...,,( 112121n

nnnn Sxxxxxxxxxv

Antipodal Preserving MapsDefine by

and by

.

11: nn RR

its restriction, also denoted by

|| n

Definition A map

f

xx )(

and the antipodal map on nS

nn SSf : or |||:| nnf is antipodal preserving if ff

).()()()( xfxfxfxf

Lemma If Example . odd,)(,: ZmzzfTTf m

cc |||:| nnf is antipodal preserving

then there exists|||:| n

mns

1m and a simplicialapproximation which is antipodal preserving.

to

Antipodal Preserving MapsProof It follows from Lebesgue’s Lemma that

such that

and Since

Partition

there exists

ff

1m

so

and for every

define so that

wstvstfwvn

mn

nmn

)(vert

nn and mn

mn

)()())(())(( wstwstvstfvstfnn

mn

mn

)(vert AAmn )(AAAv

nvertvs )(

)()( vsstvstfn

mn

and define

)).(())(( vsvs

Homology with Coefficients in a GroupDefinition If is a simplicial complex and

chain complex where for everydenote the

is the group of

is an abelian group we let );( GKCG

simplices in tensor products as

since both

of

Z

G valued functions on the set ,0q

K

);( GKCq

q .K This can be defined using GKCGKC qq )();(

)(KCq Gand are modules.

Heuristically, we can obtain );( GKCq from)(KCq by replacing integer coefficients of

simplices by elements in G and letting theboundary operator be G linear.Homology groupsare defined as usual and denoted by );( GKCq

Antipodal Preserving MapsTheorem If

is odd.

is an antipodal map then

nn SSf : or |||:| nnf

Proof It suffices to prove that)( fd

Letis an isomorphism.

,}11:{)(vert: kiv ink

22 })({);( ZzspanZH nmnn

kmkk zZC );( 2 sum of all k-simplices in

Question: prove

);();(:)( 22 ZHZHfH nnnn

mk

.0)( ns z SinceAssume that

ss

and )( kkk ccz where ).(,|,||| 1 bvertvbbc kkk Then

.1 kk zc

)()()()( nnnsnsns ddccc where nd is thesum of simplices in )( ns c containing vertex .1nv

Antipodal Preserving MapsThen apply

of simplices each which either contains

).and does not contain

where

1nd

or (contains

1nv

)()()()()()( 111 nnnsnsnsnsns ddcczcc

and this is impossible -why?

Repeat to obtain

to obtain

)]()([)()( 11 nnsnns dcdc

)()()( 111 nnnns dddc

)()( nnns ddc

nv )1( nv

is a sum

))()(()()( 1212 nnsnns dcdc )()()( 2212 nnnns dddc )()()( 1121 dddcs )()( 110 ddzs

Antipodal Preserving MapsLemma If is antipodal preserving

Proof Else

then

nm SSf :

Let is contractible

However

since

.nm

Since

nmn

mn BxxSxS }0:{ 12 and

mmnn

mn SxxxSxB }0,0:{ 132

nBfh |

)(gd

nBand

nSfg |

and nShg | it follows that

.0)( gd

nn SSg : is antipodal preserving, is odd. This contradiction concludes the proof.

Borsuk-Ulam Theorem

Theorem (BU) For every mapsuch thatthere exists

1,: nRSf nn

nSx

Proof Else the map1: nn SSg

is antipodal - contradicting the previous Lemma.

nSxxfxf

xfxfxg

,||)()(||

)()()(

defined by

).()( xfxf

Lusternik-Schnirelmann Theorem

Theorem (LS) If with

1

1

n

i in AS closed then

nn RSf :

contains a pair of antipodal points.

iA

one of the iA

Proof Since the function

defined by )),(),...,,(()( ni AxdAxdxf is continuous there exists

niiin AydAydyfyfSy ,...,1),,(),()()(

Hence if 0),( iAyd for some i then iA contains

the pair of antipodal points }.,{ yy Otherwise

1},{ nAyy and this concludes the proof.

Ham Sandwich TheoremTheorem (HS) If

nAA ,...,1 are bounded measurablesubsets of

nR there exists a hyperplane that bisects each Proof Define

nn RSf : by

nixfxM Ai i

,...,1,))(()(

where nnnn

n SxxyxyxRyxM },0:{)( 111

iA

f is continuous so the Borsuk-Ulam Theorem

implies that there exists ).()( pfpfSp n

Clearly 11 np so }0:{ 111 nnnn pypypRy

is a hyperplane that bisects each iA

Euler-Poincare FormulaDefinition Euler characteristic of a chain complex

that satisfies

qrZ qrq many finitely but allfor 0 andCq

Theorem

111

qd

qd

q CCC qq

is

Zq qq rC )1()( and Betti numbers are

Proof Smith normal form for integer matrices

)).((rank CH qq

Zq q

qC )1()(

)(rank )(rank ),(rank )(rank 1 qqqqqq BZBZr

q qq

q qq

qq BZC .)1()(rank )1()(rank )1()( 1

1

Rational Homology

is a simplicial complex with

,),( ))((rank KZq

qQQCZ

qrq QQKC ),(

Lemma If

then))((rank),( KB

qqQQCB

))((rank))((rank

),(

),(),( KBKZ

q

qq

qqQQKB

QKZQCH

K))(( KCrankr qq and ))(( KHrank qq

Proof Follows since Q is a field so that all theQ modules are vector spaces over Q

))((rank ))((rank KBKZ qqq

Hopf Trace TheoremTheorem If

is a chain map then),(),(: QKCQKC

Remark gives the Euler-Poincare Formula

K

Proof Choose bases

Theorem (HT) If is a simplicial complex and

Zq qq

Zq qq H )( trace)1( trace)1(

),(1 QKCqqqqqq

qqqcczzcc ,...,,,...,,,..., 11

111 1

for each ),( QKCq and for each basis elementw let )(w denote the coefficient of w in the

expansion in this basis for ).(w Therefore

qqq

j

qj

j

qj

j

qjq czc

111

1 )()()( trace1

Since

)()( 11 qj

qj cc so the formula follows by summing.

Lefschetz Fixed-Point TheoremTheorem If

map )( f

Theorem (LFP) If

Zq q

q fHf )( trace)1()(

Definition The Lefschetz numberof a triangulable top. space isXXf :

of a

0)( f then f has a FP.Proof Else simp. coml. 3/)(mesh , KK

1))(,(min|,|||: || mxfxdKKf Kx and simp. approx.

||||: KKs m to f and subdivision chain map ).;();(: QKCQKC m

givesFor oriented q-simplices

);();(: QKCQKCs )( fHThe chain map in K and ‘in’ )(q

.3/2))(,(3/))(),((|,||||| xsxdxfxsdx3/))(,(3/),(|| ysydyxdy so )(xs and y

do not lie in same simp. of K hence )(s so

has coef. zero in .0)( trace)()( qsqs

Lefschetz-Hopf Index Formula

manifold with isolated fixed points

n

j jpIf1

)()(

Theorem (LH) If is a map of a closedXXf :

where the indexnpp ,...,1 then

is defined

of anisolated fixed point

by

whereis defined by )()( pgdpI

)( pI

p

p)dim(,: 11 XdSSg dd

p

xxfxxfxg p )(/))(()( where we identify

points other than p with }.1||||:{ xRxB dd

a closed neighborhood of that has no fixed

Proof Henle p249 proves simple case of result in H. Hopf, U¨ber die algebraische Anzahl von Fixpunkten,Math. Z. 29 (1929), 493–524.

Also see R. F. Brown, The Lefschetz Fixed Point Theorem, 1971 and Fixed point theory, History of Topology, 1999, pp. 271–299.

Universal Coefficient Theorem for Homology Theorem (UCTH) Let ),( dC be a fee chain complex

(considered as a Z module) and G be an abelian

group. Then there exists a exact sequence0)),(()()(0 1 GCHTorGCHGCH qqq

that splits, so ).),(()()( 1 GCHTorGCHGCH qqq Proof p. 219-227 in Spanier’s Algebraic TopologyRemark Tor is a covariant functor in each argument

)ker(),(,0),(,/, mggGZTorGZTormGGGZGGZ mm

Proof p. 327-334 in Munkres’ Elem. of Alg. TopologyExample IfC is simplicial chain complex of a Kleinbottle then 0)(,)(,)( 2210 CHZZCHZCH hence

2220 );( ZZZZCH 2222221 ),()();( ZZZZTorZZZZCH

222222 ),(0);( ZZZZTorZZCH

Categorical Considerations

Tor functors. The category of left R-modules also has enough projectives. If A is a fixed right R-module, then the tensor product with A gives a right exact covariant functor on the category of left R-modules; its left derivatives are the Tor functors TorRi(A,B).Ext functors. If R is a ring, then the category of all left R-modules is an abelian category with enough injectives. If A is a fixed left R-module, then the functor Hom(A,-) is left exact, and its right derived functors are the Ext functors ExtRi(A,B).

http://en.wikipedia.org/wiki/Derived_functor

http://en.wikipedia.org/wiki/Abelian_category

The Tor functor as well as the Ext functor that arises in cohomology, are examples of derived functors in abelian categories, here are several websites to learn more:

http://en.wikipedia.org/wiki/Injective_module

http://en.wikipedia.org/wiki/Projective_module

Hopf’s Homotopy Theorem

Theorem (HH) Two maps of a sphere to itself are

homotopic if and only if they have the same degree.

Proof See p. 340, 350-354 in Dugundji’s Topology.