A bar construction in Morse-Witten homology

19 Pages·2012·0.608 MB·English

by  Emilia Lundberg

U.U.D.M. Project Report 2012:2
A bar construction in Morse-Witten homology
Emilia Lundberg
Examensarbete i matematik, 30 hp
Handledare och examinator: Tobias Ekholm
Januari 2012
Department of Mathematics
Uppsala University

Contents
1 Introduction 1
2 Preliminaries 1
3 Intersection product and higher products 3
3.1 Perturbed Morse flow trees . . . . . . . . . . . . . . . . . . . . . 3
3.2 Compactness and transversality of T(a ,b) . . . . . . . . . . . . 7
k
3.3 Higher products. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 The bar complex F(CM ) 11
∗
5 Computations for the n-sphere and products of spheres 13
5.1 The n-sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 Products of spheres . . . . . . . . . . . . . . . . . . . . . . . . . . 14
A Calculations 17
1 Introduction
In a classical paper, [2], Adams introduced a cobar construction for chain com-
plexes and used it to express the homology of the based loop space of a given
space in terms of its singular chain complex. In the present paper we present a
related construction in Morse theory. Starting from a Morse function we intro-
duce a A -structure on the Morse-Witten complex of the function by counting
∞
perturbed Morse flow trees. This leads to a Hochschild differential on the as-
sociated bar complex. Inspired by relations between the Fukaya category and
thesymplectichomologyofcotangentbundles(ormoregeneralWeinsteinman-
ifolds), see [6] and [3], we expect that the Hochschild homology of the bar
complex associated to the Morse-Witten complex on a manifold is closely re-
latedtothehomologyofthefreeloopspaceofthemanifold. Weverifythatthe
two are isomorphic for spheres of any dimension and for products of spheres by
direct calculation.
2 Preliminaries
Let X be a compact smooth Riemannian n-manifold without boundary. Let g
beaRiemannianmetriconX andf :X →RaMorsefunction,withonecritical
point of index 0 and one of index n. Also assume that (f,g) is Morse-Smale (as
is the case for (f,g) in an open dense subset).
A function is called Morse if the Hessian is non-singular at every critical
point. The index of a critical point of a Morse function is the dimension of a
maximal subspace of the tangent space at the critical point where the Hessian
is negative definite. In order to discuss the Morse-Smale condition on (f,g) we
need the following definitions.
1
Definition 2.1. For x∈X a (gradient) flow line is a smoothmap γ :R→X
x
s.t.
d
γ = −∇f ◦γ ,
dt x x
γ (0) = x.
x
A broken flow line consist of several flow lines joined at critical points.
Definition 2.2. Let a be a critical point of the Morse function f. Define the
unstable (or descending) manifold of a as
Wu(a)={x∈X| lim γ (t)=a}
x
t→−∞
and the stable (or ascending) manifold of a as
Ws(a)={x∈X| lim γ (t)=a}.
x
t→+∞
Then
dimWu(a) = λ ,
a
dimWs(a) = n−λ ,
a
where λ is the index of a.
a
Denote by M(a,b) the moduli space of flow lines from a to b modulo trans-
lation in the source with R, or equivalently the space of rigid flow lines from a
to b. Note that
M(a,b)=Wu(a)∩Ws(b)/R.
The pair (f,g) is called Morse-Smale if the stable and unstable manifolds
intersect transversally. Thus, since we assume (f,g) to be Morse-Smale then
dimM(a,b)=λ −λ −1.
a b
TheMorsedifferential∂M belowisdefinedbycountingthenumberofpoints
inthemodulispaceM(a,b)foraandbsuchthatλ −λ =1. Todothisweneed
a b
the moduli space to be compact. The following proposition and corollary are
essentiallyProposition2.35andCorollary2.36in[5]andgivesthecompactness
result. In both cases we assume that (f,g) satisfies the Morse-Smale condition.
Proposition 2.3. Every sequence (γ) ∈ M(a,b) has a subsequence (γ ) con-
n
verging to a flow line in M(a,b) or to a broken flow line in
M(a,x )×M(x ,x )×···×M(x ,b)
1 1 2 i
where x ,...,x , 1≤i≤λ −λ −1, are critical points, i.e. (γ ) converges to
1 i a b n
a broken flow line with breaks at x ,...,x .
1 i
Corollary 2.4. If λ −λ =1, then M(a,b) is a finite set of rigid flow lines.
a b
Remark 2.5. We obtain a compactification M(a,b) of M(a,b) by adding ap-
propriate broken flow lines. E.g. for a and b such that λ −λ = 2 we have
a b
that
(cid:91)
∂M(a,b)= M(a,x)×M(x,b).
x
λx=λa−1
2
Define the Morse-Witten complex CM (X,f) as the Z -vector space gener-
∗ 2
ated bycritical pointsof f andgraded by theindices ofthe criticalpoints. The
Morse differential ∂M is defined by:
(cid:88)
∂M(a)= #M(a,b)b,
b
λb=λa−1
for #M(a,b)∈Z , the number of rigid flow lines from a to b modulo 2.
2
Theorem 2.6. ∂M ◦∂M =0.
Proof.
(cid:88)
∂M ◦∂M(a) = #M(a,b)∂M(b)
λb=λa−1
(cid:88) (cid:88)
= #M(a,b)#M(b,c)c
b λc=λb−1
(cid:88)(cid:88)
= #M(a,b)×M(b,c)c
b c
(cid:88)
= #∂M(a,c)c,
λc=λa−2
which is zero since M(a,c) is a 1-dimensional compact manifold and hence the
boundary components sum to zero modulo two.
The corresponding homology HM (X) is isomorphic to the singular homol-
∗
ogy of X and so it is independent of the choice of both the Morse function and
the metric.
3 Intersection product and higher products
In this section we will define higher products (similar to the Morse differential)
whichmapsanarbitrary(finite)numberofcriticalpointstoalinearcombination
of critical points. For this purpose we want to count objects called (perturbed
Morse)flowtrees inanalogytocountingflowlinesintheMorsedifferential. The
counterpart of the Morse-Smale condition is that all relevant pertubed unstable
andstablemanifoldsoftheMorsefunctionf havetointersecttransversally. We
define the flow trees using a perturbation scheme inspired by Abouzaid [1] and
we will sketch the arguments in the first part of this section.
In the second part, we discuss transversality and compactness. We define
the higher products leading to an A -structure in the last part.
∞
3.1 Perturbed Morse flow trees
Fork ≥2defineT tobethespaceofstripswithk−1slits,seeFigure1,where
k
we make the following identification
Strip(r ,...,r )∼Strip(r +s,...,r +s),
1 k−1 1 k−1
for s∈R. Hence we may put r =0. We call the punctures at −∞ inputs and
1
the one at +∞ output. Define ∆∈T to be an infinite strip, i.e. with no slits.
1
3
0 ε 2ε 3ε kε
t
Δ Δ Δ Δ
1 2 3 k
r
1
Δ
i
r
3
Δ
Δ
j
r
2
Δ
m
r
k-1
Δ
0
s
ℝ
Figure 1: A strip ∆∈T .
k
4
Denote the infinite segments of ∆ ∈ T by ∆ for i = 0,...,k as in Figure 1
k i
and the finite parts by ∆ for i=k+1,... up to whatever number is needed to
i
label all parts. From this definition it is clear that T is diffeomorphic to Rk−2
k
for k ≥2.
ℝ
ℝ
Figure 2: Example of a broken strip.
By allowing broken strips we obtain a compactification T . These broken
k
stripscanbeviewedasthelimitofasequenceinT whereatleastone|r −r |
k i i+1
tends to infinity, an example is illustrated in Figure 2. Further, the boundary
of T is the set of all broken strips with k inputs:
k
k−2
(cid:91) (cid:91)
∂T = T × T × ···× T ,
k k1 k2 k2 k3 ki ki
i=2 k1,...,ki≥2
k=(cid:80)ij=1kj−(i−1)
whereT × T =T ×T ×{1,...,q}isthesetofbrokenstripswheretheoutput
p q q p q
oftheelementsofT isattachedtooneoftheinputsofelementsofT ,theinput
p q
isdecidedbytheelementsof{1,...,q}. ForexamplethebrokenstripinFigure
2 is an element of T × T and more specifically of T ×T ×{1}.
2 2 2 2 2
Define gluing maps for strips as
T ×T #→ρ T , (1)
k1 k2 k
for k = k +k −1 ≥ 1 and ρ ∈ (0,∞) and such that T # T → (T ,T )
1 2 k1 ρ k2 k1 k2
when ρ→∞. A gluing is illustrated below in Figure 3. For k =1 the gluing is
i
just a projection map. By the definition of gluing and by the discussion above
about broken strips it should be understood that the strips are independent of
the width.
Now, we want to define perturbation vector fields on X depending on the
stripssuchthattheperturbationdataiscompatiblewiththeperturbationdata
on the boundary. Meaning, a (possibly broken) strip in T will have some
k
perturbation data from T and some data from T wherein the unbroken strip
k k1
lies and these has to agree.
5
) )
{
ρ
,
Figure 3: Gluing of strips.
For the strip with one input we want no perturbation data. For ∆ ∈ T ,
k
k >1, we define a perturbation datum to be a choice of smooth maps
V :T ×∆ ×X →TX,
j k i
vanishing away from a bounded subset of ∆ and such that V (∆,τ,·) is a
i j
smooth vector field on X for all τ ∈ ∆ . This is defined inductively, asserting
i
the compability condition, by the following procedure:
For the strip, ∆ ∈ T , with two inputs a perturbation datum is a choice as
2
described above with j =i=0,1,2 such that V and V vanish on (−∞,−1) in
1 2
∆ and ∆ respectively and V vanish on (1,∞) in ∆ .
1 2 0 0
For k > 2 a perturbation datum for a strip near the boundary of T is
k
defined by the elements of ∂T and the gluing maps (1). For the strip ∆ ∈T
k k
where the slits are all leveled a perturbation datum is a choice as above with
j = i = 0,...,k. For all other strips in T we also have a choice of one or
k
two vector fields on the finite parts ∆ , one in each end, such that they are
i
compatible with the two cases above.
Remark 3.1. It is natural to choose the vector fields satisfying a balancing
condition at each r so that the sum of all vector fields associated to r is zero.
j j
Now we can define the (perturbed Morse) flow trees; maps T (cid:51) ∆ → X
k
which are counted in the higher products.
Definition 3.2. Let∆∈T beastripwithperturbationdata{V }l ,k ≤l≤
k i i=0
3k−4. A perturbed Morse flow tree with k inputs given by the critical points
a ,...,a and output given by the critical point b of f is a continuous map
1 k
Φ:∆→X,
whose restriction to every ∆ is a smooth map
j
φj :∆ →X,
j
such that for each t
d
φj =−∇f ◦φj +V ((t,·),φj),
ds t t i t
for j = 0,...,k and where V vanish on {(t,s)} such that s ∈ (−∞,r −1) for
i j
j =1,...,k and s∈(maxr +1,∞) for j =0.
i
6
For j > k we could have two perturbation vector fields on each ∆ , so for
j
each t we have
d
φj =−∇f ◦φj +V ((t,·),φj)+V ((t,·),φj),
ds t t i t i+1 t
whereV andV vanishon{(t,s)}suchthats∈(r +1,r )ands∈(r ,r −1)
i i+1 m k m k
respectively. Also, the φj’s should match at every r and the following should
i
hold:
lim Φ (s) = a , t∈∆ ,
t i i
s→−∞
lim Φ (s) = b.
t
s→+∞
For k =1 we define a perturbed Morse flow tree to be a gradient flow line.
a a
1 2
-∇f
-∇f
-∇f + V1 -∇f +V2
-∇f +V
0
-∇f
b
Figure 4: A pertubed Morse flow tree with inputs a ,a and output b.
1 2
A broken flow tree is, in similarity to a broken flow line, an object that
consists of several flow trees joined at critical points, an example is illustrated
in Figure 5.
For a :=a ⊗···⊗a , let T(a ,b) be the moduli space of unparametrized
k 1 k k
perturbed Morse flow trees with inputs given by a and output b. Note that
k
T(a,b) = M(a,b). Next, we will study this moduli space more carefully, but
before moving on let us define the F-degree of a .
k
Definition 3.3. For a critical point a of the Morse function f the F-degree of
a is
degF(a)=λ −n+1,
a
where n = dimX and λ is the index of a. The F-degree will also be denoted
a
by λF.
3.2 Compactness and transversality of T (a ,b)
k
In similarity to the Morse differential we want to define the higher products by
counting the elements of T(a ,b). By the following compactness and transver-
k
salityresultswemayconcludethatitisfiniteindimensionzero. Webeginwith
compactness:
7
a a
1 2
a
b 3
c
Figure 5: Example of a broken flow tree.
Theorem 3.4 (Compactness). Any sequence of flow trees in T(a ,b) has a
k
subsequence that converges to a flow tree in T(a ,b) or to a broken flow tree.
k
Hence, we obtain a compactification, T(a ,b), with boundary given by adding
k
appropriate broken flow trees.
Proof. Assume that we have a sequence of flow trees (Φ) in T(a ,b). By the
k
map π : T(a ,b) → T , which maps a flow tree to its domain with related
k k
perturbation data, we obtain a sequence of strips (π(Φ)) in T . This sequence
k
have a converging subsequence (π(Φ) ), from which we obtain a subsequence of
i
(Φ), namely (Φ ):=(π−1(π(Φ) )∩(Φ)).
j i
If, for all j > M for some M > 0, Φ is outside some neighbourhood U of
j x
all critical points x of f then Φ →Ψ for some Ψ∈T(a ,b), Figure 6. This is
j k
clear since we can choose the perturbation vector fields so that −∇f +V (cid:54)= 0
whenever −∇f (cid:54)=0.
x
1
Ux
1 x
2
Ux
2
Figure 6: For j >M, all Φ ’s are outside some neighbourhood U of all critical
j x
points x of f.
8