数学代写|COMP5045 Geometric combinatorics

(1) (a) Prove the strong Morse inequalities. That is, suppose that
$$
V: 0 \rightarrow V_n \stackrel{\partial_n}{\rightarrow} V_{n-1} \stackrel{\partial_{n-1}}{\rightarrow} V_{n-2} \stackrel{\partial_{n-2}}{\rightarrow} \cdots \stackrel{\partial_1}{\rightarrow} V_0 \rightarrow 0
$$
is a differential complex (i.e. $\partial_{i+1} \circ \partial_i=0$ for all $i$ ). Let $m_i$ denote the dimension of $V_i$, and $b_i$ the dimension of the $i^{\text {th }}$ homology $\left(=\operatorname{Ker}\left(\partial_i\right) / \operatorname{Im}\left(\partial_{i+1}\right)\right)$. Prove that for each i
$$
m_i-m_{i-1}+m_{i-2}-\cdots \pm m_0 \geq b_i-b_{i-1}+b_{i-2}-\cdots \pm b_0
$$
Make sure you see how these inequalities imply the Weak Morse Inequalities.
(b) Now prove the converse of the Morse inequalities. That is, suppose that we are given finite lists of nonnegative integers $m_0, \ldots, m_n$, and $b_0, \ldots, b_n$ which satisfy the above inequality for each $i$. Prove that there is a complex $V$ as above with $m_i=\operatorname{dim}\left(V_i\right)$ for each $i$, and such that $b_i$ is the dimension of the $i^{t h}$ homology. This shows that one cannot deduce anything stronger than the strong Morse inequalities using only the abstract existence of a complex which calculates the desired homology.

Prove that every triangulated disc is collapsible i.e. collapses to a vertex.

Triangulate a torus (more precisely, construct a simplicial complex which is homeomorphic to the torus) and find a discrete gradient field on the resulting simplicial complex with as few critical simplices as possible.

Prove that every triangulated surface has a perfect gradient vector field. That is, let $M$ be a connected simplicial complex which is homeomorphic to a compact surface. Prove that there is a gradient vector field on $M$ with precisely 1 critical vertex, 1 critical 2-simplex, and $g$ critical edges, where $g$ denotes the genus of $M$. Hint: Use the Morse inequalities to see that it is sufficient to find a discrete gradient vector field with exactly one critical vertex, and exactly one critical triangle.

One can also present discrete Morse theory using the language of functions, rather than gradient vector fields. Let $K$ be a finite simplicial complex.

A function $f: K \rightarrow \mathbb{R}$ (i.e. $f$ assigns a single real number to each simplex) is called a discrete Morse function if for each $p$-simplex $\alpha$
$$
#\left{\beta^{(p+1)}>\alpha \text { s.t. } f(\beta) \leq f(\alpha)\right} \leq 1
$$
and
$$
#\left{\gamma^{(p-1)}<\alpha \text { s.t. } f(\gamma) \geq f(\alpha)\right} \leq 1
$$
Given such a function $f$, define a set of pairs $V_f$ by declaring that ${\alpha<$ $\beta} \in V_f$ if $\alpha$ is a codimension-one face of $\beta$ and $f(\beta) \leq f(\alpha)$.
(a) Show that $V_f$ is actually a discrete vector field (i.e. that each simplex is contained in at most one pair in $V$ ).
(b) Show that $V_f$ is a gradient vector field.
(c) Show that every gradient vector field arises in this way. That is, if $V$ is a gradient vector field, then there is a discrete Morse function $f$ such that $V=V_f$.