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Results tagged with smooth-manifolds
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user 318
Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].
3
votes
Accepted
A question on parallelizability
No: any finite set $S \subset M$ can be contained in the interior of an embedded closed disc in $M$, and cutting this out gives a manifold diffeomorphic to $M - \{x\}$. So if $M - S$ were parallelisab …
- 18.2k
6
votes
Does a smooth homeomorphism of closed manifolds preserve cobordism fundamental class?
Let $X$ be an oriented $d$-manifold, and consider the homomorphism
$$\rho_i : \Omega_d(X) \to H^{4i}(X;\mathbb{Z})$$
which sends $f : M \to X$ to $f_!(p_i(TM))$, the pushforward along $f$ of the $i$th …
- 18.2k
3
votes
Cohomology classes annihilated by pullbacks
The Becker--Gottlieb transfer implies that $\pi^*$ is rationally a (split) monomorphism unless the Euler characteristic of the fibre is zero. Thus any proposed example must have this property.
- 18.2k
3
votes
Surjections on generalized homology theory
I would guess that (2') has a typo, and "into" should be "onto".
Otherwise the statement is of course not true: the degree $d$ map $S^2 \to S^2$ is injective on homology with all local coefficients, …
- 18.2k
5
votes
Is there an analog of compactified moduli spaces(/stacks) for smooth manifolds?
I have a piece of juvenilia on this topic, considering the case of zero-dimensional submanifolds. See Section 9 of
O. Randal-Williams, Embedded cobordism categories and spaces of submanifolds,
IMRN …
- 18.2k
23
votes
Accepted
Are homology spheres stably parallelisable?
Yes, they have stably trivial tangent bundles. A remark to this effect can be found on page 70 of
M. Kervaire "Smooth Homology Spheres and their Fundamental Groups"
but it is a little terse. It is e …
- 18.2k
16
votes
Accepted
Homotopy groups of Diff(X) and Homeo(X)
No, the statement about the kernel and cokernel being finite is not true.
For a closed $d$-manifold, $d \neq 4$, smoothing theory identifies the homotopy fibre of
$$B\mathrm{Diff}(M) \longrightarrow B …
- 18.2k
37
votes
Accepted
All fiber bundles over $S^2$ extendable to $\mathbb{C}P^\infty$?
No it isn't, but I had to dig quite deep to get a counterexample. Let us look at smooth $(D^7, \partial D^7)$-bundles over $S^2$, i.e $D^7 \to E \overset{\pi} \to S^2$ with an identification $\partial …
- 18.2k