nLab cup-i product

Idea

Cup-$i$ products extend cup products.

They can be used to define Steenrod squares in the same manner as ordinary cup products can be used to define the square of a cohomology class.

Definition

Given a simplicial set $X$ and $i\ge0$, we define the cup-$i$ product as the map on simplicial cochains on $X$ with coefficients in $\mathbf{Z}/2$ induced by the map on simplicial chains

$$\Delta_i: C(X,\mathbf{Z}/2) \to C(X,\mathbf{Z}/2)\otimes C(X,\mathbf{Z}/2),$$

where $\Delta_i$ evaluate on an $n$-simplex $x\in X_n$ is 0 if $i\gt n$ and

$$\sum_U d_{U^0}(x)\otimes d_{U^1}(x),$$

where $U\subset \{0,\ldots,n\}$ has cardinality $n-i$ and

$$U^k=\{u\in U\mid r(u)+k=u \pmod2\},$$

where $r(u)=\#\{v\in U\mid v\le u\}$.

Thus, we have

$$(x \cup_i y)(c) = (x\otimes y)(\Delta_i(c)).$$

Properties

We have

$$Sq^i(x) = x \cup_i x,$$

where $Sq^i$ denotes the $i$th Steenrod square.

Related concepts

• cup product

• Steenrod square

References

• Anibal M. Medina-Mardones, New formulas for cup-$i$ products and fast computation of Steenrod squares, arXiv.

• Ralph M. Kaufmann, Anibal M. Medina-Mardones, Cochain level May-Steenrod operations, arXiv.

• Anibal M. Medina-Mardones, An axiomatic characterization of Steenrod’s cup-i products, arXiv.