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Results tagged with gt.geometric-topology
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user 318
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
2
votes
Euler class of vertical tangent bundle of the surface bundle over circle
Let me not Poincare dualise, and work in cohomology.
Let me generalise the setting you have described, and consider the universal surface bundle $\pi : E \to M_g^1$ over the moduli space of surfaces w …
- 18.2k
8
votes
Accepted
Name for extension of the symplectic group
I think it is sometimes written $\operatorname{GSp}_{2g}(\mathbb{Z})$, and called the group of symplectic similitudes
- 12.9k
6
votes
Diffeomorphism groups of h-cobordant manifolds
This is regarding your second question.
In dimensions $\geq 5$, where the $s$-cobordism theorem applies, $h$-cobordisms are invertible in the following sense: if $W : M \leadsto M'$ is an $h$-cobordis …
- 18.2k
7
votes
Chromatic orientability of manifolds
If $p$ is odd then this is easy but dull. Using the truncation $k(n) \to HF_p$, a $k(n)$-orientable vector bundle is orientable in the usual sense. On the other hand, $p$-locally the Thom spectrum $MS …
- 18.2k
10
votes
Accepted
Homological stability and Waldhausen A-theory
I don't think that you can deduce homological stability of the coinvariants from the Serre spectral sequence as you suggest. But this precise situation was studied in my paper "An upper bound for the …
- 18.2k
3
votes
Accepted
Density of compactly-supported homeomorphisms
I think this is true. It suffices to prove the
Lemma. Given an orientation-preserving homeomorphism $h$ of $\mathbb{R}^n$ there is a compactly-supported homeomorphsim $h_1$ which agrees with $h$ on th …
- 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
$$B\mathrm{Diff}(M) \longrightarrow B …
- 18.2k
9
votes
Accepted
Intersection form of surface bundle over surface
Yes, such a thing exists, but I don't know an explicit example.
To see that it exists, it is clearest to me to consider the universal situation. For any $k \in \mathbb{Z}$ there is a space $\mathcal{S …
- 18.2k
4
votes
Accepted
Subdivision of closed homology manifold reference request
Using excision, and the decomposition of the star of a simplex $\sigma$ as $\sigma * \mathrm{Lk}(\sigma)$, you can easily show that your definition is equivalent to asking that
$$H_i(|K|, |K| \setminu …
- 18.2k
8
votes
Accepted
Whitney sum formula for topological Pontryagin classes
Yes. A simple argument is that $BO \to BTOP$ is a rational equivalence and an H-space map (in fact even an infinite loop map), so it follows from the Whitney sum formula for vector bundles.
Edit: The …
- 18.2k
80
votes
1
answer
3k
views
Topological cobordisms between smooth manifolds
Wall has calculated enough about the cobordism ring of oriented smooth manifolds that we know that two oriented smooth manifolds are oriented cobordant if and only if they have the same Stiefel--Whitn …
- 35.5k
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
4
votes
Immersions of manifolds with boundary (regular homotopy classes, h-principle)
There are two questions you could ask: about the space of immersions $Imm((M, \partial M), (N, \partial N))$ of $M$ in $N$ taking the boundray to the boundary, where the boundary is allowed to move, o …
- 18.2k
13
votes
Examples of Self-Maps of E8-Manifold
Such a map $f : M \to M$ of degree $d >0$ satisfies, with respect to the cup-product pairing $\langle -, - \rangle$,
$$\langle f^*(x), f^*(y) \rangle = d \langle x, y \rangle.$$
Conversely, I claim t …
- 18.2k
3
votes
Compute cohomology of flat fiber bundles - does this always work?
Let $G=\mathbb{Z}$ act on $S^1$ by an irrational rotation: this defines a flat fibre bundle $S^1 \to E \to S^1$. As $\mathbb{Z}$ is a free group this action can be deformed to the trivial action, and …
- 18.2k