algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
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
where $\Delta_i$ evaluate on an $n$-simplex $x\in X_n$ is 0 if $i\gt n$ and
where $U\subset \{0,\ldots,n\}$ has cardinality $n-i$ and
where $r(u)=\#\{v\in U\mid v\le u\}$.
Thus, we have
We have
where $Sq^i$ denotes the $i$th Steenrod square.
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.
Last revised on May 10, 2022 at 18:43:51. See the history of this page for a list of all contributions to it.