8
votes
Accepted
Integral homology classes of which no multiples admit embedded representatives with trivial normal bundle
With the trivial normal bundle condition, it's fairly easy to produce non-realizable examples using Théorème II.2 of Thom's paper. Namely, a class $z\in H_l(M^n;\mathbb{Z})$ is realizable by an ...
- 34.1k
6
votes
Accepted
How small need a perturbation be to not change the diffeomorphism type of a variety?
Let me prove $(1)$.
First of all, I guess that $f, \, g$ are homogeneous polynomials of the same degree $d$, otherwise $Z(f+ \varepsilon g)$ is not well-defined as a subvariety of $\mathbb{RP}^k$.
...
- 64.1k
3
votes
Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory
The following is relatively standard. It is almost certainly not the "deep" reason you are asking for. I think it is useful point of view though.
Suppose I have some invariant $J$ which ...
- 5,702
2
votes
Identifying a curve on a closed surface of genus 4
As mentioned in comments, your picture is not entirely accurate. But perhaps this is what you're looking for?
(Note that, if you had chosen a different gluing pattern for your once-punctured genus-...
- 23.3k
2
votes
Are these two natural cohomology classes of a manifold constructed from a 1-cochain and a group extension equal?
The general problem with this question is the definition of the Bockstein map. If $\widetilde G$ is non-abelian, then cochains with coefficients in $\widetilde G$ doesn't really make sense. The ...
- 4,238
2
votes
A Tate resolution for $\Sigma_p$ - Reference request
Your resolution is the minimal Tate complex resolving the trivial module for $\mathbb{F}_p\Sigma_p$. Each of the exterior powers of the natural permutation module is indecomposable and projective. The ...
- 4,238
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