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Results tagged with differential-topology
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user 21564
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”.
32
votes
Accepted
If two smooth manifolds are homeomorphic, then their stable tangent bundles are vector bundl...
The result you are hoping for is in fact false.
In section 9 of Microbundles: Part I, Milnor constructs an open set $U \subset \mathbb{R}^m$. With its standard smooth structure, the (stable) tangent b …
- 19.4k
23
votes
If $M$ and $N$ are closed and $M\times S^1$ is diffeomorphic to $N\times S^1$, is it true th...
The manifolds $M$ and $N$ may not even be homotopy equivalent!
In Compact Flat Riemannian Manifolds: I, Charlap showed that there are two closed flat manifolds $M$ and $N$ of the same dimension which …
- 19.4k
18
votes
Accepted
Converse to Hopf degree theorem
See the second half of the answer for a complete characterisation of closed orientable manifolds with the Hopf property.
Note that $X$ having the Hopf property is equivalent to the injectivity of $\d …
- 19.4k
15
votes
Realization problem for Betti numbers
Suppose we are given non-negative integers $b_0, b_1, \dots, b_n$ with $b_k = b_{n-k}$. Is there a closed orientable manifold $M$ with $b_i(M) = b_i$? First we need $b_0 = b_n \geq 1$. It is enough to …
- 19.4k
12
votes
Chern classes of generators of $K(S^{2n})$
Note that $K(S^{2n}) = \mathbb{Z}\times\mathbb{Z}$ has many different sets of generators and these can have different Chern classes, so the question doesn't have a well-defined answer. For one particu …
- 19.4k
11
votes
A question about the existence of spin maps
If $M$ and $N$ are spin, then every map between them is a spin map. In particular, there exist spin maps $M \to N$ of degree zero.
If $M$ is spin and $N$ is not spin, then $f : M \to N$ is a spin map …
- 19.4k
10
votes
Accepted
The maximum number of vertical independent vector fields on the tangent bundle
I will address the first version of your question (i.e. no conditions on commuting flows).
A vector bundle $E \to B$ admits $k$ linearly independent vector fields if and only if $E$ has a subbundle is …
- 19.4k
9
votes
Accepted
Index of Dirac operator and Chern character of symmetric product twisting bundle
Your first question can be answered by using the splitting principle.
If $V \to X$ is a complex vector bundle of rank two, then $c_1(S^3V) = 6c_1(V)$ and $c_2(S^3V) = 11c_1(V)^2 + 10c_2(V)$.
Pr …
- 19.4k
9
votes
Accepted
To what extent is a vector bundle on a manifold with boundary determined by its restriction ...
As I indicated in my comment, the inclusion $\iota : M_0 \to M$ is a homotopy equivalence. This can be shown using the fact that the boundary $\partial M$ has a collar neighbourhood; it then boils dow …
- 19.4k
8
votes
Every _______ $d$-manifold has an $S$-structure
In this paper, Aleksandar Milivojevic and I prove that every orientable manifold of dimension $\leq 7$ is spin$^h$. We also construct, for every $d \geq 8$, infinitely many homotopy types of closed, s …
- 19.4k
7
votes
Accepted
Stable normal bundle and immersions
This follows from obstruction theory; also see this answer.
If $E \to X$ is a rank $r$ real vector bundle over a CW complex $X$, then the obstructions to finding a nowhere-zero section lie in $H^i(X; …
- 19.4k
6
votes
Accepted
Does composition on the right by a volume-preserving diffeomorphism preserve homotopy class?
Let $X$ be a smooth, compact, orientable manifold and let $\omega$ be a choice of volume form. On $X\times X$, we have the natural volume form $\sigma = \pi_1^*\omega \wedge \pi_2^*\omega$ where $\pi_ …
- 19.4k
5
votes
Accepted
Quantitative results for stabilizing tangent bundles of homology spheres
If $E \to X$ is a rank $r$ real vector bundle, then it is classified by a map $X \to BO(r)$. The existence of an isomorphism $E \cong E_0\oplus\underline{\mathbb{R}}$ (equivalently, the existence of a …
- 19.4k
4
votes
Accepted
Under what condition is a fiber bundle cobordant to the trivial bundle?
Here are three special cases where it is always true:
Sphere bundles of vector bundles. If $E \to B$ is a vector bundle and $S(E) \to B$ is the associated sphere bundle, then $S(E)$ bounds the disc b …
- 19.4k
4
votes
Atiyah-Bott-Shapiro generalization to $U(n) \to ({Spin(2n) \times U(1)})/{\mathbf{Z}/4}$ for...
Let $\omega = e_1e_2\dots e_{2n-1}e_{2n}$.
For $n > 1$, the center of $Spin(2n)$ is $Z(Spin(2n)) = \{\pm 1, \pm\omega\}$. Note that $\omega^2 = (-1)^n$, so
$$Z(Spin(2n)) = \begin{cases}
\langle -1, \ …
- 19.4k