7 votes
Accepted

Restriction vs. multiplication by $n$ in Tate cohomology

Let $G = \Bbb Z/4$, acting on the Gaussian integers $M = \Bbb Z[i]$ via multiplication by $i$. The transfer $M_G \to M^G$ is given by multiplication by $1 + i + (-1) + (-i) = 0$, so $$H^{-1}(G,M) = ...
  • 48.1k
7 votes

Dual surfaces of a first cohomology class of a 3-manifold

Some things are known in the non-orientable case (in orientable 3-manifolds). Bredon and Wood work out the genus in lens spaces $L(2n,q)$ (and some other manifolds) in their paper Non-orientable ...
5 votes

How to define cohomology of algebraic structures?

There is a tremendous amount of abstract formalism answering this question in various levels of generality depending on what you want to do. I'll pick one in the middle: the machinery of derived ...
5 votes
Accepted

Formula for the Euler characteristic of a local system on $\mathbb{P}^1$

The answer to your question at the end is negative. In fact, $h^2(D, j_*F)= h^2(D, j_* F)=0$. In fact, the cohomology of a sufficiently small disc around a point in any complex variety, with ...
  • 119k
5 votes

Dual surfaces of a first cohomology class of a 3-manifold

$L(4,1)$ is a counterexample to your conjecture, taking $\alpha$ to be the nontrivial element of $H^1(L;\Bbb Z/2)$. Notice that this element squares to zero (the square is the same as the Bockstein, ...
  • 8,415
2 votes

Fundamental class of topological compact surfaces

Firstly, if you start with one triangle then there is a unique sign for any neighbouring triangle such that the common boundaries add up to zero. Thus you can start at one triangle and distribute ...
  • 475

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