The cup-product tag has no wiki summary.

**3**

votes

**1**answer

233 views

### Computations of cup products in Serre's Local Fields

I have been reading the appendix in Serre's Local fields, to do with explicit computations of cup products (pg 176), but I'm stuck on one bit of lemma 4. It goes as follows
Let B be a $G$-module, $u: ...

**4**

votes

**2**answers

654 views

### Why are cup-i products and Steenrod Squares often (always?) unary?

One way to define the Steenrod Operations is to use the cup-i product, as in Mosher and Tangora's book. It basically says, given the chain complex from mod-2 homology $C_\ast$, define
$D_0 : ...

**2**

votes

**0**answers

210 views

### Defining the cup product in Ext using a Kunneth formula

I want to make a Kunneth product of sorts on Ext. In particular, letting $C_\*$ be a $R$-free resolution for $k$ over a $k$-hopf algebra $R$, elements in $Ext_R(k,k)$ are represented by maps in ...

**2**

votes

**2**answers

522 views

### Non-vanishing of cup product in cohomology

Let $X$ be a smooth projective variety over $\mathbb{C}$, and let $\alpha \in H^{2k}(X)$ be an algebraic cohomology class. Let us fix an $l$ such that $H^l(X) \neq 0$ and $H^{l+2k}(X) \neq 0$.
The ...

**9**

votes

**4**answers

1k views

### Techniques for computing cup products in singular cohomology

Suppose that we are given a CW complex X in terms of the cells and the gluing maps. My understanding is that computing the cup product of the singular cohomology ring from this information is a ...

**8**

votes

**2**answers

401 views

### On-the-nose commutative cup product $\Longrightarrow$ characteristic $0$?

I was discussing with a student today the nature of the non-commutativity of cup-product at the level of cochains. In trying to explain what happens, I came up with the following statement.
...

**2**

votes

**3**answers

684 views

### Another group cohomology cup product question

I am wondering if there is a way to see the cup product, in some cases, without using cochain complexes. The situation I am interested in is the following:
Let $G=F/R$ be a finitely presented group ...