nLab generalized cup product

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 $\cup_1$.

Related concepts

• cup product

• sequence operad

References

• James E. McClure, Jeffrey H. Smith,

  Multivariable cochain operations and little $n$-cubes (arXiv:math/0106024)