# Questions tagged [cup-product]

The tag has no usage guidance.

29 questions
Filter by
Sorted by
Tagged with
155 views

### Generalisation of Hirsch formula for the associativity of Steenrod's higher $\cup_2$ product with $\cup_1$ and cup products

For $f$, $g$ and $h$ cochains, the Hirsch formula is given as $$(f\cup g)\cup_1 h=f\cup (g\cup_1 h)+(-1)^{q(r-1)}(f\cup_1 h)\cup g.$$ Is there a more general formula that relates the associativity of ...
169 views

### 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 ...
231 views

### 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 $\operatorname{char} F$ ...
181 views

### 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 ...
107 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 ...
104 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 ...
1k views

320 views