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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”.
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 $ …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
11
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 …
- 18.5k
18
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 …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
6
votes
Searching for an unabridged proof of "The Basic Theorem of Morse Theory"
My recollection is that Milnor's proof gives exactly what you are asking. In fact, see the remark on the bottom of page 17 of his book.
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
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)\, .
$$
Further, $D(f)$ is a prop …
- 18.5k
8
votes
1
answer
1k
views
The free smooth path space on a manifold
Let $M$ be a closed, smooth manifold and let $PM$ be the space of unbased piecewise smooth paths $[0,1] \to M$. Then restricting a path to its boundary gives a map
$$
PM \to M \times M .
$$
Question …
- 18.5k
5
votes
Accepted
Is $\partial \Gamma\hookrightarrow \Gamma$ a Serre cofibration?
Regarding the first question: assume $M$ is compact. According to the "fundamental theorem" of Morse theory, there is a filtration
$$
M_{-1} \subset M_0\subset \cdots \subset M_m = M
$$
where $M_{-1} …
- 18.5k
15
votes
Parallelizability of the Milnor's exotic spheres in dimension 7
The following is just an expansion of Johannes' last paragraph.
I went to Adams' paper where he attributes to Dold the statement that $S^n$ parallelizable implies $S^n$ is an $H$-space. No referenc …
- 18.5k
4
votes
A map of spaces implementing the Pontryagin Thom collapse map? (collapse maps in families)
The situation is somewhat easier to describe if one replaces the embeddings of the closed codimension $(m-n)$manifold $M$ with the embeddings of the total space of a disk bundle of a rank $(m-n)$-vect …
- 18.5k
18
votes
1
answer
567
views
Local homology of a space of unitary matrices
Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$). Let
$$
{\cal D} \subset U(n)
$$
denote the subspace of those matrices having
a non-trivial $(+1)$-eigenspace.
Backgroun …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
9
votes
0
answers
345
views
History of the definition of smooth manifold with boundary
I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it …
- 18.5k
6
votes
0
answers
123
views
On the weak homotopy type of a differentiable (Chen) space
Suppose that $M$ is a differentiable space in the sense of Chen (cf. https://ncatlab.org/nlab/show/Chen+space ).
Assume that $M$ also has the structure of a topological space and that the two struct …
- 18.5k