Contents

# Contents

## Idea

Steenrod’s generalized cup products generalize the usual simplicial cup products on simplicial cochains.

Taken together, all generalized cup products organize into the sequence operad. This operad is an E-infinity operad and it acts on simplicial cochains, turning them into an E-infinity algebra.

## Definition

First, we define an operation on simplicial chains that decomposes simplices in a manner similar to how the usual simplicial cup product decomposes a simplex into the tensor product of a simplex consisting of the initial $(p+1)$ vertices and a simplex consisting of the last $(q+1)$ vertices, with $p+q$ being the dimension of the original simplex.

###### Definition

(Definition 2.10(a) in McClure-Smith 01.) Given a surjection $f\colon\{1,\ldots,m\}\to\{1,\ldots,k\}$ and a simplex $\sigma\colon\Delta^p\to X$, define

$\sigma[f]\in(C(X))^{\otimes k}$

as

$\sigma[f]=\sum_A (-1)^{\epsilon(f,A)}\bigotimes_{1\le i\le k}\sigma\left(\coprod_{f(j)=i}A_j\right).$

The variable $A$ is indexed over all overlapping partitions of $\{0,\ldots,p\}$ with $m$ parts.

Here

$\epsilon(f,A)=\sum_{j\lt j',f(j)\gt f(j')}dim(A_j)dim(A_{j'})+\sum_{1\le j\le m}dim(A_j)(\tau_f(j)-f(j)),$

(Definition 2.9 in [Cochain]) where

$\tau_f(j)=card\{j'\in\{1,\ldots,m\}\mid f(j')\lt f(j) or f(j')=f(j) and j'\le j\}$

(Definition 2.7 in [Cochain]). Also, $dim(B)=card(B)-1$ denotes the dimension of the simplex with the set of vertices $B$ (Notation 2.6 in [Cochain]).

We are now ready to define generalized cup products.

###### Definition

(See Definition 2.10(b) in [Cochain].) Given a surjection $f\colon \{1,\ldots,m\}\to\{1,\ldots,k\}$, the natural transformation

$\langle f\rangle=\cup_f\colon (C^*(X))^{\otimes k}\to C^*(X)$

is defined by

$\langle f\rangle(x_1\otimes\cdots\otimes x_k)(\sigma)=(-1)^{m-k}(x_1\otimes\cdots\otimes x_k)(\sigma[f]).$

###### Remark

(See Lemma 2.12, Definition 2.13, and Definition 2.14 in [Cochain].) If $f(l)=f(l+1)$ for some $1\le l\lt m$, then $\langle f\rangle=0$. Thus, in the sequence operad such degenerate operations must be modded out. The remaining operations form a basis of the sequence operad.

###### Remark

(See Remark 2.11 in [Cochain].) If $k=2$, we recover the traditional cup-$i$ products $\cup_i$ defined by Steenrod as follows. Define $f\colon\{1,\ldots,m\}\to\{1,2\}$ to be the function with alternating values 1, 2, 1, 2, … Now $\langle f\rangle=\cup_{m-2}$. In particular for $m=2$ (the sequence 12) we recover the cup product and for $m=3$ (the sequence 121) we recover $\cup_1$.