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 ...

- 17.8k

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 ...

- 109k

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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