Semana 4

The ingredients of the complex introduced by Forman [32] are conceptually close to those of the traditional Morse complex. One begins with a topological CW complex, on which is defined some suitable function f. From f is derived a collection of critical cells (critical points) and integral curves (connecting orbits). New cells are formed by combining critical cells with the integral curves that descend from them, and the resulting family is organized into the structure of a complex via incidence relations.

Let us make this description precise. Posit a regular topological CW complexX, and let X(p) denote the associated family ofp-dimensional cells. We write σ(p) whenσ has

dimension pand τ > σ when σ lies in the boundary ofτ. A function

f :X→R

is adiscrete Morse function if

#{τ(p+1)> σ:f(τ)≤f(σ)} ≤1 (10.2.1)

#{υ(p−1) > σ:f(υ)≥f(σ)} ≤1. (10.2.2)

A cell σ(p) iscritical with indexp if equality fails in both cases, that is if

#{τ(p+1)> σ:f(τ)≤f(σ)}= 0

#{υ(p−1) > σ:f(υ)≥f(σ)}= 0.

Thegradient vector field off is the family

∇f =

n

(σ(p), τ(p+1)) :σ < τ, f(τ)≤f(σ)

o

.

It can be shown that

#{(σ, τ)∈ ∇f :σ=υ or τ =υ} ≤1 (10.2.3)

for any υ∈X, so∇f is a partial matching on the incidence relation<.

(10.2.3) holds with R in place of∇f. Agradient path on R is a sequence of form

α₀(p), β₀(p+1), α(₁p), β₁(p+1), . . . , β(_rp+1), α(_rp₊₁) (10.2.4)

where

(αi, βi)∈V αi+1 < βi and αi 6=αi+1

for all i∈ {0, . . . , r}.

Much like the Morse complex, the discrete Morse complex has a basis indexed by critical cells and a differential expressible as a sum of values determined by integral curves. One assigns an orientation to each cell of X, and defines the multiplicity of a gradient path γ by m(γ) = k−1 Y p=0 −h∂βp, αpih∂βp, αp+1i.

Let Cf denote the free abelian group generated by the critical cells of X, graded by dimension, and let Γ(β, α) denote the set of gradient paths that run from a maximal face of β toα. The discrete Morse boundary operator is the degree−1 map on ˜∂:Cf →Cf

defined by ˜ ∂β= X critical α(p) cα,βα, (10.2.5) wherecα,β =PΓ(β,α)m(γ).

Theorem 10.2.1 (Forman [32]). The pair (Cf_,∂˜_{) is a complex, andH(C}f₎∼_=H(X,

The discrete Morse complex is a highly versatile algebraic tool, with natural applications in topology, combinatorics, and algebraic geometry. The relation between smooth and discrete Morse theory has been refined by a number of results over the past two decades, e.g. [34], where it was shown that every smooth Morse complex may be realized as a discrete one, via triangulation. Given the breadth and depth of the correspondence between smooth and discrete regimes, it is reasonable to consider wether an analog to the Hodge-theoretic approach of Witten might exist for cellular spaces, as well. This is indeed the case, as shown by Forman [33].

In the spectral setting, one begins with a suitably well-behaved CW complex X, equipped with a discrete Morse function f. For each real numbertone defines an operator

etf _{on C}

∗(X,R) by

etfσ=etf(σ)σ

forσ ∈X. To the associated boundary operator one assigns a family of differentials

∂t=etf∂e−tf.

to which correspond a t-parametrized family of chain complexes

(C, ∂t) : 0−→Cn ∂t −→Cn−1 ∂t −→ · · · ∂t −→C0 −→0

with associated Laplacians

Here∂_t∗ denotes the linear adjoint to∂t with respect to the inner product onC∗(X,R) for

which basisX is orthonormal. The operator ∆(t) is symmetric, hence diagonalizable, and for each λ∈Rwe denote theλ-eigenspace of ∆(t) by

Eλ_p(t) ={c∈Cp : ∆(t)c=λc}.

Since ∂t∆(t) = ∆(t)∂t, operator∂tpreserves eigenspaces. For each λ∈Rone therefore

has a differential complex

Eλ(t) : 0−→E_nλ(t) ∂t

−→E_nλ₋₁(t) ∂t

−→ · · · ∂t

−→E₀λ(t)−→0.

Forman defines a well-behaved class of Morse functions (flat Morse functions) and shows that for flat f

∆(t)→     0 0 0 D    

as t→+∞, whereD is a block submatrix indexed by the set{σ1, . . . , σk}of critical cells

of f, and Dhas form

diag(aσ1, . . . , aσk),

for somea∈(Z>0∪ {+∞}){σ1,...,σk}.

Ast→ ∞, therefore, the spectrum of ∆(t) separates into a portion close to zero and a portion close to one. Fixing arbitrary 0< ε <1, we thus define the Witten complex of f

to be

writing W(t) : 0−→Wn(t) ∂t −→Wn−1(t) ∂t −→ · · · ∂t −→W0(t)−→0.

for the natural grading induced by dimension.

Let π(t) denote orthogonal projection from C(X,R) to Wp(t), and for each critical

p-cellσ put

gσ(t) =π(t)σ.

One has gσ(t) =σ+O(e−tc) for some positive constantc. The gσ form a basis of Wp(t)

whentis sufficiently large, but not an orthonormal one in general. If we define Gto be the square matrix indexed by critical p-cells such that

Gσ1,σ2 =hgσ1, gσ2i,

however, and

hσ =G−1/2gσ,

then the hσ do form an orthonormal basis. Forman’s observation is that the matrix

representation of ∂t with respect to this basis tends to that of the discrete Morse complex.

Theorem 10.2.2 (Forman [33]). Suppose that f is a flat Morse function. Then for any critical σ(p) and τ(p) there exists a constant c >0 such that

h∂thτ, hσi=et(f(σ)−f(τ)) h h∂τ, σ˜ i+O(e−tc) i , where ∂˜is as in (10.2.5).