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Results tagged with differential-topology
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user 8032
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
3
votes
Accepted
Homotopy between sections
Not in general.
Suppose $f: S^1\times T \to S^1$ is the projection, where
$f$ is the first factor projection and $T = S^1 \times S^1$ is the torus.
Then a section amounts to a map $S^1 \to T$ and the …
- 18.9k
8
votes
How can I construct a closed manifold from a finite CW complex?
More generally, suppose $n \le m$ are non-negative integers,
$X$ is a CW complex of dimension $\le n$, $M$ is a non-empty, closed $m$-manifold,
and $X$ and $M$ have the same homotopy type.
It is well …
- 18.9k
9
votes
Accepted
Atiyah duality without reference to an embedding
Here is another short construction which is much simpler and just takes a few lines.
Let $M$ be a closed $n$-manifold. Consider the diagonal $M \to M \times M$. It is an embedding. Take its Pontryagi …
- 18.9k
7
votes
Atiyah duality without reference to an embedding
Assume $M$ is a closed, smooth manifold of dimension $n$. Let $\tau^+$ be the fiberwise one point compactification of its tangent bundle. This is a fiberwise $n$-spherical fibration equipped with a p …
- 18.9k
1
vote
$G$-equivariant intersection theory using differential topology?
You may want to take a look at
Klein, J.R., Williams, B.
Homotopical intersection theory, II: equivariance.
Math. Z. 264(2010),849–880.
An arXiv version appears here:
https://arxiv.org/abs/0803.0017
I …
- 18.9k
14
votes
Unstable manifolds of a Morse function give a CW complex
(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps.
I will retract this for now. I do recall being told this, but I am not
aware at this point in time where the gaps …
- 18.9k
19
votes
Accepted
Realizing cohomology classes by submanifolds
Your question is just a reformulation of what Thom did, so the answer is always yes.
Since the Stokes map from de~Rham cohomology to singular cohomology (with real coefficients) is an isomorphism, yo …
- 18.9k
15
votes
Parallelizability of the Milnor's exotic spheres in dimension 7
The following is just an expansion of Johannes' last paragraph.
I went to Adams' paper where he attributes to Dold the statement that $S^n$ parallelizable implies $S^n$ is an $H$-space. No referenc …
- 18.9k
4
votes
A map of spaces implementing the Pontryagin Thom collapse map? (collapse maps in families)
The situation is somewhat easier to describe if one replaces the embeddings of the closed codimension $(m-n)$manifold $M$ with the embeddings of the total space of a disk bundle of a rank $(m-n)$-vect …
- 18.9k
5
votes
Obstruction Theory for Vector Bundles and Connections
Correction: the definition below is wrong. It isn't true that a 1-flat reduction is the same as a flat reduction. One also needs to require that the map $Z \to BG$ factors through $BG^\delta$, where $ …
- 18.9k
10
votes
Accepted
Non-zero homotopy/homology in diffeomorphism groups
Here is a very naive approach: choose a basepoint in the manifold (call it $M$). Then evaluation at the basepoint gives a map
$$
\text{Diff}(M) \to M
$$
and so cohomology classes on $M$ pull back to o …
- 18.9k
5
votes
Accepted
Is $\partial \Gamma\hookrightarrow \Gamma$ a Serre cofibration?
Regarding the first question: assume $M$ is compact. According to the "fundamental theorem" of Morse theory, there is a filtration
$$
M_{-1} \subset M_0\subset \cdots \subset M_m = M
$$
where $M_{-1} …
- 18.9k
2
votes
Ehresmann fibration theorem for manifolds with boundary
Let $D(M)$ be the boundary of $M \times [0,1]$ (by smoothing corners, this can be understood as smooth). Then $f: M \to N$ induces a smooth map
$$
D(f): D(M) \to D(N)\, .
$$
Further, $D(f)$ is a prop …
- 18.9k
3
votes
Accepted
Vector field pull back from embedding
At each point $x\in M$ the differential $df_x: T_x M \to T_{f(x)}N$ is a monomorphism. However, if $X$ is a vector field on $N$ the vector $X_{f(x)}$ need not be in the image of $df_x$. Hence to assoc …
- 18.9k
1
vote
What does it mean that homotopy is generic?
"Generic" usually refers to open and dense.
Assume $M$ is a closed smooth manifold. Let $$C^\infty(M)$$ be the space of all smooth real valued functions. Topologize this with respect to the Whitney …
- 18.9k