Contents

Contents

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.

References

Last revised on May 10, 2022 at 18:43:51. See the history of this page for a list of all contributions to it.