All Questions
Tagged with at.algebraic-topology differential-topology
503
questions
16
votes
0
answers
404
views
Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
- 467
6
votes
0
answers
206
views
"Inclusion" between higher categories of framed bordisms?
Let $\mathrm{Bord}_n$ be the bordism $(\infty, n)$-category of unoriented manifolds.
It can be viewed as an $(\infty, n+1)$-category whose $n+1$-morphisms are equivalences.
If $n$ is large enough, ...
- 890
9
votes
0
answers
138
views
degree 1 maps for bordism homology
Let $f\colon X \to Y$ be a degree 1 map between closed oriented manifolds. Then the induced homomorphism between the homology groups is surjective up to torsion.
Can one say something similar about (...
1
vote
0
answers
140
views
Higher dimensional Seifert surfaces and link numbers of higher knots
In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots.
Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...
- 493
5
votes
0
answers
127
views
Division of fibration by $\Sigma_{n}$ gives Serre fibration
This is related to a question posted on StackExchange: https://math.stackexchange.com/questions/4776877/left-divisor-of-a-fibration-by-compact-lie-group-is-a-fibration. The question there had received ...
- 151
1
vote
0
answers
122
views
Can we construct a general counterexample to support the weak whitney embedding theorm?
The weak Whitney embedding theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m > 2n$.
...
- 109
2
votes
0
answers
133
views
Explicit S-duality map
$\DeclareMathOperator{\Th}{Th}$
The Thom space of a closed manifold $M$ ($\Th(M)$) is the $S$-dual to $M_+ (=M \cup \{pt\})$. Let $M$ be embedded in $\mathbf R^{n+k}$. I found the duality map from $S^...
13
votes
1
answer
353
views
Realizing integral homology classes on non-orientable manifolds by embedded orientable submanifolds
Let $M^m$ denote a compact, non-orientable smooth manifold and $\nu$ an integral homology class of dimension $n$. I am interested in understanding the representability of $\nu$ by embedded, orientable ...
- 467
8
votes
1
answer
201
views
Connectivity of the space of transverse vector fields
Suppose we have a smooth, closed manifold $M$ of dimension $n$ and connectivity $k$. What can we say about the connectivity of the space of all tangent vector fields on $M$ that are transverse to the ...
- 2,052
0
votes
0
answers
52
views
Role of basins of attraction in the Morse decomposition
Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by
$$\dot{x}=F(x(t))$$
An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of ...
- 247
1
vote
0
answers
57
views
Given a proper submersion $f: X \setminus F_0 \to D \setminus 0$ which extends to $X \to D$, are cycles on $X$ which run around $0$ boundaries in $X$?
I have a proper map of complex manifolds
$$f: X \to D,$$
where $D \subset \mathbb C$ is the unit disc. By assumption, $f$ has connected fibers, is smooth over $D \setminus 0$, and a smooth fiber $F$ ...
- 1,141
2
votes
0
answers
83
views
Turning cocycles in cobordism into an inclusion or a fibering
By the classical Pontryagin-Thom construction, we know that the cobordism group $\Omega^n_U(X)$ is represented by cocyles
$$ M\hookrightarrow X\times \mathbb{R}^{2k}\rightarrow X,$$
where $M$ is a ...
- 171
5
votes
0
answers
133
views
Geometric interpretation of pairing between bordism and cobordism
In page 448 of these notes, a pairing between bordism and cobordism
$$\langle \ ,\ \rangle: U^m(X)\otimes U_n(X)\rightarrow \Omega^U_{n-m}$$
is defined as follows. Assume $x\in U^m$ is represented by $...
- 171
11
votes
1
answer
326
views
Embedded 2-tori in $S^1\times S^4$
I am interested in understanding the smooth isotopy class of embedded 2-tori in $S^1\times S^4$. Is it true that every two homotopic embedded 2-tori in $S^1\times S^4$ are smoothly isotopic? It would ...
- 129
8
votes
0
answers
144
views
Generators for unstable cobordism
I am looking for explicit descriptions of generators of some low-dimensional unstable cobordism groups. For example, $\mathbb CP^2$ embeds into $\mathbb R^7$ by a result of Haefliger. Because it has ...
- 6,701