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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 10th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItex## 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 $i>n$ 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 $r(u)=\#\{v\in U\mid v\le u\}$. ## References * [[Anibal M. Medina-Mardones]], _New formulas for cup-$i$ products and fast computation of Steenrod squares_, [arXiv](arXiv:2105.08025). * [[Ralph M. Kaufmann]], [[Anibal M. Medina-Mardones]], _Cochain level May-Steenrod operations_, [arXiv](https://arxiv.org/abs/2010.02571). * [[Anibal M. Medina-Mardones]], _An axiomatic characterization of Steenrod's cup-i products_, [arXiv](https://arxiv.org/abs/1810.06505). <a href="https://ncatlab.org/nlab/revision/cup-i+product/1">v1</a>, <a href="https://ncatlab.org/nlab/show/cup-i+product">current</a>

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

   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.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 10th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added context menu and table of contents. Are you going to create the page *[[Anibal M. Medina-Mardones]]*? Please do, otherwise the broken links in the entry look bad. <a href="https://ncatlab.org/nlab/revision/diff/cup-i+product/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/cup-i+product/2">v2</a>, <a href="https://ncatlab.org/nlab/show/cup-i+product">current</a>

   I have added context menu and table of contents.

   Are you going to create the page Anibal M. Medina-Mardones? Please do, otherwise the broken links in the entry look bad.

   diff, v2, current