MATEMATIQKI VESNIK UDK

originalni nauqni rad

research pap er

GENERALIZED COHOMOLOGY AND MORSE COMPLEX

Darko Milinkovic and Zoran Petrovic

Abstract In this pap er we analyse Morse complex in the framework of generalized

cohomology theory This analysis was suggested in

Intro duction

It is well known that using a Morse function on a compact smo oth manifold

one can recover integral cohomology of this manifold It is interesting to see what

can b e said ab out generalized homology Of course since a Morse function actually

determines appropriate CW complex homotopy equivalent to the given manifold

one knows a priori that it should give information ab out generalized homology as

well but one wants to knowhow generalized cohomology can b e computed using

a Morse function

We give rst some background on sp ectral sequences Here we follow the

illuminating After that we remind the treatment as in which we nd most

reader ab out the basic notions of generalized homology and give a sp ectral sequence

which one uses to compute generalized homology or cohomology of a given ltered

spacethe AtiyahHirzebruch sp ectral sequence

We apply these techniques in the case when we havea Morse function on a

given compact manifold Weknow from general considerations that this sp ectral

sequence converges toward generalized cohomology of our manifold but our

task is to show what is the second term of our sp ectral sequence and how it is

related to the Morse function in question Wealsogive an example whichshows

how one computes using this sequence For the reason of simplicity when wedeal

with actual computations we only use generalized homology with Z co ecients

although more general results remain valid

AMS Subject Classication B T

Keywords and phrases Morse complex generalized homology sp ectral sequence

This work was partially supp orted by Ministry of Science and Environmental Protection of Re public of Serbia Pro ject

D Milinkovic Z Petrovic

At the end we p onder on higher dierentials in this sp ectral sequence in order

to better understand geometry of the situation which ordinary homology leaves

b ehind

Wewould like to express our thanks to the anonymous referees for their valu

able suggestions whichhave improved the presentation in this pap er

Sp ectral sequences and generalized cohomology

This material is fairly standard For the readers convenience we supply some

basic denitions and facts

Lo ok at the following diagram consisting of graded ab elian groups and homo

morphisms

i i

i i i

A w A w w A A

p p p p

j j j

k k k

  

E E E

p p p

We are using here homology typ e notation and we assume that each triangle

A A E A

p p p p

from this diagram is a long exact sequence This is what Boardman see calls

unrol ledexactcouple Since we express a bias toward homology i and j have degree

while k has degree Of course all constructions work as well for the case when

k has degree

r 

r fold iterate of iwe get the following diagram If we denote by i

r 

i i i

A A w A w w A

pr  p pr p

j j

k k

 

E E

pr p

We use this diagram to intro duce the following notation

 r  r

k Imi A A Z

pr p

p

r  r

j Keri A A B

p pr 

p

r r r

B Z E

p p p

r r

The subgroup Z is known as the group of r cycles and B as the group of r

p p

r

b oundaries of E The group E is a comp onent of the E term of the sp ectral

p r

p

r r

sequence whichwe construct Of course E E The following inclusions hold



   

E Z Z Im j Kerk Z B B B

p

p p p p p p

r

Dierential d in this sp ectral sequence is induced by the additive relation

r  

e relations see Namely we would like to take an j i k for additiv

element of E apply k then lift this result r times using i and nally p

Generalized cohomology and Morse complex

apply j to what we got lo ok at the previous diagram We cannot do this for

all elements and even if we could the result would only b e dened up to a certain

subgroupthat is how one intro duces the groups of r cycles and r b oundaries One

r r r

can check that in this way d maps E into E In conclusion some diagram

p pr

r r r 

chase shows that H E d E and we really have a sp ectral sequence

We can also intro duce the following groups

 r  r   

Z Z B B E Z B

p p p p p p p

r r



If we are able to relate this E to some ltered group G than we have solved

p

the convergence problem for this sp ectral sequence Namely supp ose that G is



ltered by subgroups F F G in such a way that E F F so we are

p p p p

p

using a ltration in which F F increasing ltration Then wesay that our

p p

sp ectral sequence converges to G and we denote this by

r

E G

where r corresp onds to the last term in our sp ectral sequence for which we are



able to nd a decent presentation usually it is the second term

Next we discuss generalized cohomology Basic reference for this material

is or As weknow a generalized cohomology theory satises all Eilenb erg

Steenro d axioms except for the dimension axiom namely homology of a p oint need

not b e trivial in nonzero dimensions This is rather imp ortant for what follows We

will b e working with multiplicative homology theory therefore homology of a space



e

h X isamodule over the ring of co ecients h h pt or h S in the case of

   

reduced theory Let us just mention a few examples of generalized cohomology

theories to illustrate their dierence from ordinary homology

Example Complex K theory The ring of co ecients KU for the homol





ogy theory is the ring of nite Laurent series Z t t with generator t in degree

This generator corresp onds to the isomorphism arising from Bott p erio dicity

Example Complex cob ordism Here the ring of co ecients for the homolo

gy theory is MU Z x x where the p olynomial generators x are in degree

  i

i



Example Morava K theories The ring of co ecients here is K n

 n

F v v where the degree of the generator v is p Wewould liketo

p n n

n

pointoutthat wehave innitely many Morava K theoriesfor dierent n and p

and they are all p erio dic theories

When we are given a ltered space X and a multiplicative homology theory

h as usual h denotes b oth the theory in question and its ring of co ecients we

 

can construct a sp ectral sequence as follows Denote by X the pth space in the

p

ltration of X Then wehave the following unrolled exact couple

i i i

h X w h X w h X w h X

 p  p  p  p

uv uv uv

u u u

u u u

k k k

j j j

  

h X X h X X h X X

 p p  p p  p p

D Milinkovic Z Petrovic

The resulting sp ectral sequence is the AtiyahHirzebruch sp ectral sequence It con

verges to the h X ltration on this group is induced by the ltration on X



The rst dierential d of this sp ectral sequence is given by the comp osition jk



j

k

d h X X h X h X X

  p p  p  p p

We examine a version of this in sections that follow

Morse functions and the resulting sp ectral sequences

R Let us rst recall a few general facts ab out Morse functions Let f M

b e a Morse function on a smo oth closed manifold M Wedenoteby Crit f the

p

set of its critical p oints of index p For two critical p oints x x of f we denote



by Mx x the set of all R M satisfying



d

rf t x as t



dt

This equation denes oneparameter ow on M Note that

t

u s

Mx x W x W x

 

where

s

W x f x M j lim xx g

 t 

t

u

W x f x M j lim xx g

t

t

are the stable and unstable manifolds of the negative gradientow F or a generic

choice of Riemannian metric used to dene the gradient rf or equivalentlyfor

a generic choice of Morse function f the intersection ab oveistransverse Morse

Smale condition Hence the set Mx x is a smo oth manifold of dimension



mx mx where mx denotes the Morse index of a critical p oint x



The group R acts on Mx x via s We denote the orbit space



of this action by

c

Mx x Mx x R

 

c

The manifolds Mx x can b e given a coherentorientation see for more



c

details ObviouslydimMx x dim Mx x It is known that when

 

c

mx mx the zero dimensional manifold Mx x is compact and

 

thus a nite set

Let f M R be a Morse function on a smo oth closed manifold M We

denote by Crit f the set of its critical p oints of index p and dene

p

C f free ab elian group generated by Crit f

p p

The graded ab elian groups C f can be turned into the chain complex in the



following way For a generator x Crit f we dene

p

X

nx y y x

p

y Crit f 

p

Generalized cohomology and Morse complex

and extend it to the map

C f C f

p p p

Here nx y is the signed with resp ect to the orientation number of points in

c

zero dimensional compact manifold Mx y Of course H f Ker Im

p p p

It turns out that H f H M where H M denotes the singular homology of

  

M One also derives from this chain complex the wellknown Morse inequalities

This is the classical p oint of view

As we see in the classical case only the onedimensional mo duli spaces of

tra jectories are imp ortant Thus we can say that ordinary Morse homology

algebraizes the onedimensional mo duli spaces In the case of generalized homology

mo duli spaces of any dimension play their role and the generalized homology is

computed using a sp ectral sequence as is shown b elow That construction therefore

give an algebraization of the higher mo duli spaces Another way to algebraize them

is given by Barraud and Cornea

In order to get to our sp ectral sequence we follow the idea that go es backto

Smale Milnor Witten and Flo er Let f M R b e a Morse

function suchthatf xmxforevery critical p oint x see Chapter in for





We use this the pro of of existence of such function Let M f p

p

ltration for the sp ectral sequence we are going to construct Using notation from

the previous section we put

A h M E h M M

p  p p  p p

Deforming along the gradient lines of f we get the homotopy equivalence

S

u

M M W x M M

p p p p

xCrit f 

p

The rst dierential in our sp ectral sequence is

d h M M h M h M M

  p p  p  p p

namely the connecting homomorphism for the triple M M M By exci

p p p

sion wehave the following isomorphism

M

u

h M M h W x M M

 p p  p p

xCrit f 

p

so in order to determine d wehave to determine its xy comp onent where x stands



or for a critical p oint of index p while y stands for a critical p ointofindexp F

that let

u u  s s 

S xW x f p S xW x f p

u u

The group h W x M M is generated by the class S x susp ension

 p p

of the unstable sphere Namely

u u u

e e

h W x M M h W x M M h S x

 p p  p p 

u

Under the connecting homomorphism of the pair W x M M this class

p p



u 

M and consider the comp osition go es to S x Let L f p

p

u

h S x h L h M h M M

   p  p p

u

W y M M h

p p 

y Crit f 

p

D Milinkovic Z Petrovic

If we pro ceed with the pro jection to the y comp onent and use the homotopy equiv

alence

u s

W y M M L L n S y

p p

provided by the gradientow we see that in order to determine our dierential we

u

e

have to nd out where do es the generator of h S x go under the homomor



u s

phism h S x h L L n S y induced by the inclusion

 

u

S x L

n s

ywetake a lo ok at the homomorphism h L L n S y Using Poincare dualit

n u

h S x To get what we need we invoke the Thom isomorphism theo

s

rem Namely after restriction to some tubular neighborhood of S y in L we

s

may assume that L is a vector bundle over S y and that the generator of

n s u

h L L n S y is the Thom class Let D S x be the generator of

n u  u

h S x and let D S x where h Then



 u u u

h S xi hD S x S xi

s

But is the PoincaredualofS y and hence

u s u  u s

S x S y S x h h S xi S y



From and we conclude that the rst dierential of our sp ectral sequence

is the dierential of the complex

M

C f h hCrit f i

 q p

pq 

where h hCrit f i is a free h mo dule over the set Crit f Therefore wehave

q p q p

proved the following theorem

Theorem Let M be a closed manifold let f be a selfindexing Morse

function on M and h a multiplicative homology theory There exists a spectral



sequenceconverging to h M with the second term given by



f

E H M h

q

pq p

f

where by H M h we denote a homology calculated from the fol lowing Morse

q

p

complex

M

C f h hCrit f i

 q p

pq 

where the dierential is given on generators by

X

s u

dvx v S y S xy

y Crit f 

p

for v h and x Crit f

 p

In case this sp ectral sequence collapses at its second term we get that

L

f

h M H M h This happ ens in the case of ordinary homology

n q

p

pq n

Generalized cohomology and Morse complex

The reason is simpleall higher dierentials b ecome zero since they either origi

nate from the trivial group or target the trivial group

Note that may b e written using the Conley index as

M

C f h N L

  x x

xCrit f 



where N L is the Conleys Index pair of an isolated compact invariantset fxg

x x

see for more details concerning the Conley index

Critical p oints here are a sp ecial case of a more general notion of the Morse

decomp osition see for the denition asso ciated to a ow on a compact manifold

This suggests that the Theorem might b e generalized to the Conley index setting

although it is not immediately clear whether the dierentials will b e so explicitly

tractable in the most general case

Examples and higher dierentials

We illustrate our construction with the following simple example

Let us dene a generalized homology theory as follows



h X H S X

 

 

Let us consider the case X S If wecho ose the height function on S as our

only two singular p oints one of index and one of index Morse function we get

There are only two tra jectories Our sp ectral sequence in this case reduces to a

  

complex in which all dierentials are zero Therefore h S H S H S

  

 

Here H S denotes a copyof H S shifted by one dimension We therefore

 

 

recover ordinary homology of torus S S but by using a simpler complex than

in the classical case

Let us at the end saya few words ab out higher dierentials in this sp ectral

sequence Naturallyitisvery dicult in the general case to deal with them But

thecaseinwhich there are no critical p oints at a certain level it is p ossible to say

something general ab out them

Supp ose that weha ve a manifold and a Morse function on it such that there

are no critical p oints of o dd index

n

Complex pro jective spaces CP provide examples of such manifolds Namely

n n 

if weseeCP as the quotient space S S wemay dene the following function

n

X

f z z k jz j

 n k

k 

It turns out that this is a Morse function which has n critical points with

indices n We know that in the case of ordinary homology we get a

trivial complex and ordinary homology is isomorphic to this complex In the case

this do es not happ en We cannot recover generalized of generalized homology

homology immediately We can only conclude that d and moveto d What



D Milinkovic Z Petrovic

we get for d in this case is the expression very muchliketheonewehave had for



s u

d The only but very imp ortant dierence is that in this case S y S xisnot

a nite number of points but is homotopy equivalent to a compact onedimensional



manifold namely to a nite numb er of copies of S Of course it dep ends on the

generalized homology one uses to pro ceed further with calculations

In general in case there are no critical p oints of index say p q

several dierentials b ecome trivial and one has to deal with the case in which

s u

S y S x is a compact manifold of dimension higher than two Its fundamen

tal class in the generalized theory then b ecomes part of an expression for higher

dierentials

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J F Adams Stable Homotopy and Generalised Homology Chicago Lectures in Math

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retic Methods in Nonlinear Analysis and in Symplectic TopologyP Biran O Cornea and

F Lalonde Eds NATO Science Series Springer pp

J M Boardman Conditional ly convergent spectral sequencesContemp Math

C Conley Isolated invariant sets and the Morse index CBMS Regional Conference in

Mathematics Vol American Mathematical So cietyProvidence RI

E Dyer Cohomology Theories W A Benjamin

A Flo er Wittens complex and innite dimensional Morse theory J Di Geom

D McDu D Salamon Introduction to Symplectic Topology Oxford Mathematical Mono

graphs The Clarendon Press Oxford University Press New York

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D Salamon Connected simple systems and the Conley Index of isolated invariant sets

Trans Amer Math So c

M Schwarz Morse Homology Birkhauser Basel

S Smale The generalized Poincarec onjecture in dimensions greater than Ann of Math

S Smale On gradient dynamical systems Ann of Math

S Smale On the structure of manifolds Amer J of Math

G W Whitehead Elements of Homotopy Theory SpringerVerlag New York

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received in revised form

Matematicki fakultet Studentski trg IV Beograd Serbia

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