Hipótesis Específico (1)

To formulate the Leray-Serre spectral sequence for a non-simply-connected base it is con- venient to use the formalism of local coefficients, introduced for singular and cellular homology by Steenrod [Ste43]. See also Section 5.3 in [McC01a]. Our discussion follows Section 7.2 in [Oan], who also discusses Morse homology with local coefficients.

Definition 8.3. Let R be a ring. A system of local coefficients on a topological space B is given by a collection of R-modules {Mb}_b∈B, called fiber at b, together with a family

of isomorphisms

Φα : Mα(0)→α(1)

α∈P (B),

called parallel transport along α, where P (B) := C0([0, 1], B) denotes the set of continuous paths in B, s.t.

1. if b denotes the constant path equal to b ∈ B, then Φb = idMb,

2. if α, β ∈ P (B) are homotopic with fixed end points, then Φα = Φβ,

3. if α, β ∈ P (B) satisfy α(1) = β(0), then

Φα·β = Φβ ◦ Φα

where α · β ∈ P (B) denotes the product path of α and β.

If B is not path-connected we assume in addition that all the Mb, b ∈ B, are isomorphic.

Given a local system of coefficients all the fibers are isomorphic as R-modules. It is customary to talk about a local system of coefficients with (typical) fiber M, where M = Mb for some b ∈ B.

Any R-module M defines a trivial system of local coefficients via Mb := M for all b ∈ B

and Φα := idM for all α ∈ P (B). Obviously any system of local coefficients on a simply-

connected base B is isomorphic with a trivial local system of coefficients.

Given a local system of coefficients S on a topological space B it is possible to define singular homology with local coefficients H∗(B; S), see [Ste43] or [McC01a], generalizing

singular homology with constant coefficients.

Next we define Morse homology with coefficients in a local system S, extending Section 7.2.1 in [Oan03], whose treatment carries over to Morse homology on Hilbert manifolds without difficulties. For simplicity we neglect orientations and work over Z2. Our basic

reference for Morse homology on Hilbert manifolds is [AM04], which we interpret in terms of [Sch93].

Let (B, g) be a complete Riemannian Hilbert manifold1, endowed with a smooth Morse function f , s.t.

1. f is bounded from below

2. every critical point of f has finite Morse index, 3. f satisfies the Palais-Smale condition,

4. −∇f satisfies the Morse-Smale condition up to order 2. Let furthermore a system of local coefficients

S ={Mb}_b∈B,Φα : Mα(0)→α(1)

α∈P (B)



of _Z2-vector spaces on B be given.

Denote by Critkf the set of critical points of f with Morse index k. Define for all k ∈ N

a _Z2-vector space

Ck(f ; S) :=

M

x∈Critkf

Mx

1_{With some work Morse homology with local coefficients can even be defined for the most general}

setting we are aware of, namely a C1-Morse vector field on a Banach manifold admitting a C1−Lyapunov function which only satisfies the Palais-Smale condition up to order 0 as in Section 2.3 of [AM04].

and linear maps ∂k : Ck(f ; S) → Ck−1(f ; S) by (8.3) ∂k(m) := X y∈Critk−1f X [γ]∈ cMx,y Φγ˜(m), m ∈ Mx, x ∈ Critkf,

where Mx,y denotes the set of all smooth maps γ : R → B satisfying γ0 = −∇f ◦ γ

together with the asymptotic conditions lim

s→−∞γ(s) = x and s→+∞lim γ(s) = y.

The set Mx,y is either empty or a one-dimensional manifold, in which case the action

(8.4) R × Mx,y → Mx,y, (τ, γ) 7→ γ(· + τ )

is free and the set

c

Mx,y := Mx,y

 R

consists of a finite number of points. Due to the asymptotic conditions any γ ∈ Mx,y

can also be reparameterized to give a map ˜γ : [0, 1] → B with ˜γ(0) = x and ˜γ(1) = y. If γ1 and γ2 differ by the R-action introduced in (8.4) the paths ˜γ1 and ˜γ2 turn out to be

homotopic with fixed endpoints, in which case condition 1 in Definition 8.3 ensures that Φ˜γ1 = Φ˜γ2, implying that the summand in (8.3) is well-defined. Furthermore as f satisfies

the Palais-Smale property the number of non-zero terms appearing in (8.3) is finite. Thus we can show the Morse homology theorem with local coefficients.

Proposition 8.4. Let f be a Morse function on the complete Riemannian Hilbert mani- fold (B, g) satisfying the assumptions 1 − 4 above. Furthermore let

S ={Mb}b∈B,Φα : Mα(0)→α(1)

α∈P (B)



denote a given system of local coefficients of _Z2-vector spaces on B. Then the operators

(∂k)k∈N (8.3) satisfy the identity

(8.5) ∂k◦ ∂k−1 = 0, k ∈ N,

turning (C∗(f ; S), ∂) into a chain complex whose homology H∗(f ; S) is called Morse ho-

mology of f with local coefficients in S. Furthermore

(8.6) H∗(f ; S) ∼= H∗(B; S),

where the second term denotes singular homology with local coefficients in S.

Proof: The proof of the identity (8.5) rests upon the fact that if x and z are critical points of f with m(x) − m(z) = 2 and a non-compact connected component of cMx,z has

two ends corresponding to pairs (u1, v1) and (u2, v2) with

for critical points yi with m(yi) = m(x) − 1 (i = 1, 2), then the Gluing Theorem 7.12

shows that the path ˜u−1₂ ◦ ˜v₂−1◦ ˜v1 ◦ ˜u1, connecting x with itself, is homotopic with fixed

endpoints to the constant path equal to x, implying the identity Φv˜1 ◦ Φu˜1 = Φ˜v2 ◦ Φu˜2,

proving (8.5).

Setting up the Morse complex with local coefficients as in Section 2.7 in [AM04], to- gether with the argument of Section 2.8 of [AM04] gives (8.6) for any sublevel set Ba = f−1_{((−∞, a)) of f for a ∈ R. A direct limit argument as in Section 2.9 of [AM04] allows} to conclude (8.6).

In Section 7.2.1 of [Oan] Oancea gives a proof of Proposition 8.4 for Morse cohomology and a compact base B. He works with a negative pseudo-gradient field Y for f which he assumes to equal a non-degenerate quadratic form in the vicinity of the critical points of f to work will the cell decomposition of B induced by the flow of Y . Working with the cellular filtration of [AM04] renders this assumption superfluous.

Apparently the definition of Morse homology with local coefficients, see (8.3), does not require the full data contained in a local system. In fact only the fibers over the critical points of the Morse function are taken into account. Similarly only the paths induced by Morse trajectories connecting critical points of Morse index difference 1 enter. Axioma- tizing the proof of Proposition 8.4 leads to the notion of a local Morse system associated to (f, g).

Definition 8.5. Let f be a Morse function on the complete Riemannian Hilbert manifold (B, g) satisfying the assumptions 1 − 4 above. A local Morse system associated to (f, g) with typical fiber M, an R-module, is given by a family {Mx}x∈Crit(f )of R−modules which

are isomorphic with M together with a collection of isomorphisms {Φα} with

Φα : Mx→ My,

where x, y ∈ Crit(f ) with m(x) − m(y) = 1 and α ∈ Mx,y, satisfying the following

properties.

1. Suppose x and y are critical points of f with m(x) − m(y) = 1 and α ∈ Mx,y. Then

Φα = Φα(·+τ ) for all τ ∈ R.

2. Suppose that x and z are critical points of f with m(x) − m(z) = 2 and a non- compact connected component of cMx,z has two ends corresponding to pairs (α1, β1)

and (α2, β2) with

(αi, βi) ∈ Mx,yi × Myi,z, i = 1, 2

for critical points yi with m(yi) = m(x) − 1 (i = 1, 2). Then the identity

Φα1 ◦ Φβ1 = Φα2 ◦ Φβ2

Ideas more general than Definition 8.5 enable the construction of characteristic classes via Morse homology, see [Voi06], and the reconstruction of vector bundles from so-called transport functions, see Section 2.3 in [Voi], leading to a Morse K-Theory.

We conclude the following analog of Proposition 8.4.

Proposition 8.6. Let f be a Morse function on the complete Riemannian Hilbert mani- fold (B, g) satisfying the assumptions 1 − 4 above. Furthermore let

S ={Mx}x∈Critf , {Φu}



denote a given local Morse system associated to (f, g) of _Z2-vector spaces. Then the linear

operators ∂k: Ck(f ; S) → Ck−1(f ; S), where Ck(f ; S) := M x∈Critkf Mx, defined by (8.7) ∂k(m) := X y∈Critk−1f X [γ]∈ cMx,y Φγ(m), m ∈ Mx, x ∈ Critkf,

are well-defined and satisfy the identities

(8.8) ∂k◦ ∂k−1 = 0, k ∈ N,

turning (C∗(f ; S), ∂) into a chain complex whose homology H∗(f ; S) is called Morse ho-

mology of f with local Morse coefficients in S.

Thus a local Morse systems associated to (f, g) turns out to be the natural setting for Morse homology associated to (f, g) with local coefficients. In Section 8.3 we will construct a natural local Morse system.

By the gluing theorem any local system of coefficients gives rise to a local Morse system of coefficients. Conversely any system of Morse coefficients can be extended to a unique local system of coefficients, as described in Sections 7.2 and 7.3 in [Oan]. In fact Oancea provides the arguments for a fixed system of Morse coefficients, using only the axioms of Definition 8.5. Again his arguments rely on the cell decomposition induced by the flow of Y. See the discussion after the proof of Proposition 8.4.

8.3

The local Morse system of coefficients for the Leray-