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 ...

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 $\mathbb{RP}^k$.
...

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 ...

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-...

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 ...

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 ...

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