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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”.
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; …
- 19.3k
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 …
- 19.3k
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 …
- 19.3k
29
votes
1
answer
889
views
Is there an explicit description of a cobordism between $\mathbb{CP}^n$ and $\mathbb{RP}^n\t...
With a little bit of work, one can show that $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$ have the same Stiefel-Whitney numbers, so by a theorem of Thom, they are (unorientedly) cobordant.
…
- 19.3k
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_ …
- 19.3k
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 …
- 19.3k
2
votes
Existence of a spin map from a standard sphere to any closed Riemaninan manifold with nonneg...
For now, let $M$ be any closed oriented $n$-dimensional manifold, and $f : S^m \to M$ a smooth map.
First suppose $k \geq 1$.
Since $m = n + 4k > 2$, we have $H^2(S^m; \mathbb{Z}_2) = 0$, and hence $f …
- 19.3k
15
votes
1
answer
823
views
Are there non-smoothable homotopy/homology spheres?
A homotopy sphere is a topological $n$-manifold $M$ which is homotopy equivalent to $S^n$.
A homology sphere is a topological $n$-manifold $M$ such that $H_i(M) \cong H_i(S^n)$ for all $i$.
Note, by …
- 19.3k
9
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 …
- 19.3k
3
votes
A topological consequence of Riemann-Roch in the almost complex case
I just wanted to point out how this question is related to (spin${}^c$) Dirac operators and their indicies since this was alluded to in the comments to the question.
Let $(M, g)$ be an $2n$-dimension …
- 19.3k
21
votes
1
answer
1k
views
Are homology spheres stably parallelisable?
A homology sphere is a closed smooth $n$-dimensional manifold with the same homology groups as $S^n$. Igor Belegradek's answer to a previous question of mine shows that the smoothness hypothesis is no …
- 19.3k
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 …
- 19.3k
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 …
- 19.3k
3
votes
Topology of the blowup of a surface at a point (connected sum)
A much stronger statement is true. The following is from Huybrechts' Complex Geometry (note however that the emphasis is mine):
Let $x \in X$ be a point in a complex manifold $X$. Then the blow-up …
- 19.3k
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 …
- 19.3k