All Questions
17
questions
3
votes
0
answers
175
views
Morse theory for isotopies of codimension 1 submanifolds and generalized bigon moves
I seem to have managed to convince myself of the validity of a certain result. If it does indeed hold (modulo non-catastrophic adjustments) I could really use a reference. Otherwise, I would also be ...
- 151
1
vote
0
answers
189
views
Existence of Morse function on suspension
Let $X$ be a smooth simply connected compact manifold of dimension $n$ with boundary. Let $Y$ be a smooth compact manifold of dimension $n+1$ without boundary such that $H_{i+1}(Y)=H_{i}(X)$(reduced ...
- 61
10
votes
0
answers
180
views
A minimal $\mathbb Z/2$-invariant Morse function on $U(n)$
Consider the group $U(2n)$ of unitary matrices. This has two standard and important homomorphic involutions. The most famous one $A \mapsto \overline A$ is complex conjugation, with fixed-set $O(2n)$, ...
- 9,473
9
votes
1
answer
263
views
Smooth Morse function from Forman's discrete Morse function
Let $M$ be a smooth manifold and $K$ a triangulation of $M$, so $K$ is a regular CW-complex and in particular a simplicial complex. Assume that $M$ is compact so $K$ is finite. Let $f\colon K \to \...
- 909
15
votes
3
answers
2k
views
Unstable manifolds of a Morse function give a CW complex
A coauthor of mine and I want to use the following innocent looking statement in a forthcoming paper:
Statement. Let $M^{2n}$ be a compact manifold and let $f$ be a Morse function with critical ...
- 28.8k
1
vote
0
answers
120
views
Topological invariants of a certain "stratified" manifold, with pieces of different "dimensions"
Disclaimer: I don't fully understand what I'm talking about in the question below. I'm still trying to figure out the right question to ask. Quotations and question marks in brackets mean that I'm not ...
- 6,824
6
votes
0
answers
317
views
Elementary questions about Morse-Bott functions
Let $M$ be a manifold, $F$ be a Morse-Bott function, $c$ be a critical level, and $M_c$ be the corresponding critical submanifold. Let us assume that $M_c$ is connected, and the index of $M_c$ is ...
- 14.1k
14
votes
0
answers
270
views
Homotopy type of spaces of functions with few critical points
Given a closed manifold $M$ and an integer $k\geq 0$, let $G_k(M)$ denote the space of smooth functions $f:M\to\mathbb R$ with at most $k$ critical points.
To what extend has the topology of the ...
- 18.5k
11
votes
1
answer
546
views
Relation between Morse Theory and integration against Euler Characteristic
I'm studying Robert Ghrist papers on integration against Euler Characteristic. I am particularly interested in the relation with Morse Theory. I am trying to understand the proof of Theorem 25.1 (page ...
- 909
6
votes
2
answers
805
views
Can a Morse function define a unique structure on a closed manifold?
I was thinking about the doubt that if $M$ and $N$ are closed manifold and if there exists two Morse function $f$ and $g$ respectively on $M$ and $N$ with the following property that they both have ...
- 3,711
1
vote
0
answers
102
views
Global topological equivalence of Morse functions
Two Morse functions $f$ and $g$ are called topologicaly equivalent if there are diffeomorphism $h$ of the source and orientational-preserving diffeomorphism $k$ of the target such that $f=k\circ g\...
- 75
1
vote
0
answers
133
views
restriction to the boundary in Morse theory
Given a compact manifold with boundary $(M,\partial M)$. There is a natural pullback map
$$ H^*(M) \to H^*(\partial M) $$
I'm wondering if there is a reference that:
1) constructs this map in ...
- 1,331
9
votes
1
answer
853
views
A description of cellular boundary maps in terms of a Morse function
I'm writing a paper on classical Morse Theory and I'm interested in applying Morse functions to the computation of homology groups of a compact manifold $M$. The standard way in which this is done is ...
- 319
4
votes
0
answers
573
views
Topological version of two results in smooth Morse theory
Morse theory is generally presented in the DIFF category. However, there is a version of Morse theory in TOP (see the post Morse theory in TOP and PL categories? for references).
It is well known ...
- 2,106
15
votes
1
answer
814
views
Nonisotopic homotopy equivalent Morse functions
One can cut a manifold up along the critical levels of a Morse function and deduce something about the topology. In particular the critical points (and the connecting gradient flowlines) define a ...
- 6,975