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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”.
1
vote
Tubular neighborhoods of chains
Here's an approach which might work (I'm not sure about the correctness of this.)
1) Assume $M$ is closed. Choose a triangulation $T$ of $M$.
If the support of $c$ is contained inside the $p$-skelet …
- 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 …
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
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
17
votes
Accepted
Finite-dimensionality for de Rham cohomology
We can restrict your problem to the case of open manifolds. It turns out that "finite dimensional" integral singular homology (i.e., finitely generated in each degree) is almost the same thing as the …
- 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
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
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
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
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
12
votes
Accepted
Definition of Pontrjagin Classes
The odd Chern classes of the complexified bundle are of order 2 and are determined by the Stiefel-Whitney classes of the original real bundle $\xi$ by the formula
$$
c_{2k+1}(\xi\otimes \Bbb C) = \bet …
- 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
12
votes
Accepted
How does the Framed Function Theorem simplify Cerf Theory?
The framed function theorem tells you that up to "contractible choice" a compact manifold admits a framed function: i.e., a function as you prescribe. Furthermore, a framed function is supposed to giv …
- 18.9k
10
votes
Can we decompose Diff(MxN)?
When $N$ is the circle there's a sort of answer. In fact there's a whole chapter in the book
Burghelea, Dan; Lashof, Richard; Rothenberg, Melvin: Groups of automorphisms of manifolds. With an append …
- 18.9k
6
votes
Stratification of smooth maps from R^n to R?
It looks to me that what you are really interested in is the Thom-Boardman stratification of the function space. For that I would recommend the well-written, Stable Mappings and Their
Singularities b …
- 18.9k