# Questions tagged [cup-product]

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22
questions

**2**

votes

**1**answer

101 views

### Find $a$ satisfying $x \cup_1 y = \delta a$ when $x,y \in Z^2(G,\mathbb{Z}_2)$

Let $G$ be a finite group. Let $x,y \in Z^2(G,\mathbb{Z}_2)$ be 2-cocycles. Find $a \in C^2(G,\mathbb{Z}_2)$ such that
\begin{align}
x \cup_1 y = \delta a.
\end{align}
Is there a general solution? Is ...

**1**

vote

**0**answers

116 views

### To see that the fundamental class of a local complete intersection is independent of choice of regular sequence

In SGA 4½ ‘Cycle,’ Grothendieck defines (among other things) the fundamental class of a local complete intersection $Y\subset X$ ($X$ simply a noetherian scheme) of codimension $c$ locally as the cup-...

**1**

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**0**answers

79 views

### Application of cup-product for Leibniz cohomology

In 1995, Loday introduced a cup-product operation on the graded cohomology of Leibniz algebra and showed that the cup-product operation satisfies the graded Zinbiel relation.
My question is how this ...

**3**

votes

**1**answer

193 views

### Cup product in Tate Cohomology Ring

Let $G$ be a finite cyclic group of order $p$. The tate cohomology groups $\hat{H}^*(G, \mathbb{F}_p)$ are defined using a complete resolution of $\mathbb{F}$ as $\mathbb{F}_pG$-module.
There is a ...

**4**

votes

**1**answer

301 views

### How to construct cup-product in a general site?

Let $\mathcal{C}$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $\mathcal{C}$. Do we have cup-product as follows?
...

**9**

votes

**1**answer

430 views

### Use of Steenrod's higher cup product and the graded-commutativity

In Steenrod's ``Products of Cocycles and Extensions of Mappings (1947),'' which derives [Theorem 5.1]
$$
\delta(u\cup_{i} v)=(-1)^{p+q-i}u\cup_{i-1}v+(-1)^{pq+p+q}v\cup_{i-1}u+\delta u\cup_{i}v+(-...

**8**

votes

**1**answer

420 views

### Cup products in the Mayer-Vietoris sequence

Let $(X;U,V)$ be an excisive triad and consider the corresponding part of the Mayer-Vietoris sequence $H^{\bullet-1}(U\cap V)\stackrel{\delta^*}{\to} H^\bullet(X)\to H^\bullet(U)\oplus H^\bullet(V)$. ...

**4**

votes

**1**answer

94 views

### Equivalence of finiteness of $spliG$ and periodicity isomorphisms being induced by cup product

I am trying to prove that the following are equivalent for a group $G$ with periodic cohomology with period $q$ after $k$ steps:
$(i)\ spliG<\infty$ (where $spliG$ is the supremum of injective ...

**7**

votes

**2**answers

494 views

### Parabolic cohomology of modular groups and cup-products

I am stuck with a technical question concerning parabolic cohomology of modular groups and cup-products on them. Basically, I am trying to understand the appendix about cohomology of Hida's book "...

**1**

vote

**1**answer

104 views

### Cartan Formula for Steenrod square on cocycles

Let $x_n,y_n,\cdots$ be cocycles in $Z^n(X,\mathbb{Z}_2)$ (not cohomology classes in $H^n(X,\mathbb{Z}_2)$).
Let $Sq^k(x)\equiv x_n \cup_{n-k} x_n$ be the Steenrod square (This definition is valid for ...

**12**

votes

**1**answer

387 views

### Calculations of cup products in Bredon cohomology

Let $G$ be a finite group. In [1], Bredon defines an equivariant cohomology theory for $G$-CW complexes $H^*_G(X;M)$. The coefficients are taken in modules over the orbit category of $G$, that is, ...

**13**

votes

**0**answers

332 views

### What is the cup-product structure like on a hyperbolic 5-manifold?

Let $X$ be a hyperbolic 5-manifold. Can there be any class in $H^2(X;\mathbf{Z})$ that is torsion and whose square in $H^4(X;\mathbf{Z})$ is not zero?
For example, are there hyperbolic 5-manifolds ...

**4**

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**0**answers

198 views

### Pairing in Group Cohomology [closed]

I am following Ararat Babakhanian's Cohomological Methods in Group theory.
Let $A,B,C$ be $G$ modules then we have a $G$ module structre on $\text{Hom}_{\mathbb{Z}}(B,C)$ with $$\sigma.f(x)=\sigma(f\...

**4**

votes

**1**answer

176 views

### Formula relating the cup product in dimensions n and n+1

Let's will write $K_n$ for the Eilenberg-MacLane space $K(\mathbb{Z},n)$. I remind that $K_n$ is equivalent to the loop space of $K_{n+1}$.
Let’s consider the map $\smallsmile:K_n\times K_m \to K_{n+...

**3**

votes

**1**answer

291 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: ...

**5**

votes

**2**answers

1k 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 : C_\ast\...

**5**

votes

**0**answers

549 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 $Hom_R(...

**3**

votes

**2**answers

651 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 ...

**16**

votes

**4**answers

3k 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 non-...

**11**

votes

**2**answers

564 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.
...

**3**

votes

**3**answers

830 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 ...

**35**

votes

**3**answers

4k views

### Calculating cup products using cellular cohomology

Most algebraic topology books (for instance, Hatcher) contain a recipe for computing cup products in singular or simplicial homology. In other words, given two explicit singular or simplicial ...