Questions tagged [cup-product]

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Cup-product in cohomology and Hopf algebra

Let $X$ be a manifold and let its cohomology $H^*(X;\mathbb{Z})=\bigoplus_{q=0}^\infty H^q(X;\mathbb{Z})$ be a module of finite type without $p^2$-torsion for any prime integer $p$. Assume that on ...
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5 votes
1 answer
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Examples of a group $G$ and an $F$-representation $V$ where $\cup:H^1(G,F)\otimes H^1(G,V)\to H^2(G,V)$ is injective

Let $G$ be a group and $F$ a field. I am particularly interested in the case where $G$ is a uniform lattice in a Lie group and $F=\mathbb{F}_2$, or in finite groups $G$ where $\operatorname{char} F$ ...
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4 votes
0 answers
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Multi-variable cohomology operations

Intuitively, cohomology operations are ways to locally compute a cocycle $\alpha\in H^i(X, G)$ from any cocycle $\beta\in H^j(X, H)$. Formally, they are in one-to-one correspondence with homotopy ...
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4 votes
0 answers
98 views

Is there a local simplicial formula for the Steenrod squares which commutes with the derivative on cochain level?

There is a well-known formula for the cup product of an $i$-cochain $A$ and $j$-cochain $B$ in simplicial homology given by $$(A\cup B)(0\ldots i+j) = A(0\ldots i) B(0\ldots j)\;.$$ This formula ...
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2 votes
1 answer
95 views

Cohomology ring on non-simplicial complex

Cohomology ring and cup product can be defined on simplicial complex (ie a triangulation of a manifold). Can we define cohomology ring and cup product on a more general complex? In particulate, I am ...
10 votes
1 answer
850 views

What's the cohomology ring structure of a blow-up?

Let $X$ be a compact Kähler manifold, with $j_Z: Z\hookrightarrow X$ a submanifold of complex codimension $r$, $\tau: \widetilde{X} \to X$ the blow-up of $X$ along $Z$, with exceptional divisor $j: E \...
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2 votes
1 answer
113 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
142 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-...
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1 vote
0 answers
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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
392 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 ...
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4 votes
1 answer
426 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? ...
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9 votes
1 answer
720 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+(-...
9 votes
1 answer
652 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 $H^{\bullet-1}(U\cap V)\stackrel{\delta^*}{\to} H^\bullet(X)\to H^\bullet(U)\oplus H^\bullet(V)$. ...
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4 votes
1 answer
97 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
705 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 "...
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3 votes
1 answer
165 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
470 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, ...
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13 votes
0 answers
351 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 votes
0 answers
251 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\...
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4 votes
1 answer
179 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
309 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
2k 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
595 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
742 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 ...
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18 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-...
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11 votes
2 answers
620 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
957 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 ...
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39 votes
4 answers
6k 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 ...
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