5
$\begingroup$

One way to define the Steenrod Operations is to use the cup-i product, as in Mosher and Tangora's book. It basically says, given the chain complex from mod-2 homology $C_\ast$, define

$D_0 : C_\ast\to C_\ast\otimes C_\ast$,

so that the cup product is given (on cocycles)

$(u\cup v)(\sigma) = (u\otimes v)(D_0\sigma)$

and then for higher $i$, define $D_i$ so that

$D_{i-1}+\rho D_{i-1} = D_i\partial + \partial D_i$

where $\rho$ is the flipping map. Then the $\cup_i$ product is just

$(u\cup_i u)(\sigma) = (u\otimes u)(D_i\sigma)$

And then define for $[u]\in H^n$

$Sq^{2n-i}([u]) = [u\cup_{i}u]$

This definition seems perfectly well-defined as a binary operation, and yet wherever I've seen it done it has only even been used as a unary operation.

Is there a reason why this is the case, why either the product is undefined as a binary product or not useful as a binary product or just too hard to use?
Is this a dumb question?

Thanks, -Joseph

$\endgroup$
3
  • 3
    $\begingroup$ I think it does get used as a binary product. For instance, $u \cup_i v$ measures how far $u \cup_{i-1} v$ is from being commutative. So all the $\cup_i$ together are telling you information about the classical $\cup$ product (which is $\cup_0$) in the same way that all the levels of $A_\infty$ together give you a homotopy associative product. I think the main reason to move from $\cup_i$ to $Sq^i$ is that for the application Mosher and Tangora want (division algebras) they care about operations on cohomology, i.e. from $H^*$ to $H^*$. $\endgroup$ Aug 10, 2012 at 0:16
  • 3
    $\begingroup$ The cup_i-product of two closed chains may be nonclosed, if I remember correctly. $\endgroup$ Aug 10, 2012 at 9:46
  • 1
    $\begingroup$ While the $\cup_i$'s are useful, they may see less attention due to the fact that they do not satisfy a nice list of properties, but the steenrod operations do. $\endgroup$ Aug 12, 2012 at 19:42

2 Answers 2

12
$\begingroup$

I use $\cup_i$ products as binary products in my work on the algebraic theory of surgery

http://www.maths.ed.ac.uk/~aar/papers/ats2.pdf

They give the higher symmetry properties {$\phi_s|s \geq 0$} of the Poincare duality chain equivalence $$\phi_0=[M] \cap - : C(M)^{m-*} \to C(M)$$ of an $m$-dimensional manifold $M$, with $$d\phi_s+\phi_sd^*+\phi_{s-1}+\phi_{s-1}^*=~0~(\phi_{-1}=0)$$ up to sign.

$\endgroup$
8
$\begingroup$

Under suitable hypotheses, Gugenheim and I use the binary $\cup_1$ and especially the fact that it is a graded derivation (Hirsch formula) as the key to giving a calculation of $H^*(G/H)$ as the torsion product over $H^*(BG)$ of $R$ and $H^*(BH)$, where coefficients are taken in a suitable commutative ring $R$. This even works for suitable $H$-spaces. See http://www.math.uchicago.edu/~may/BOOKS/GugMay.pdf and http://www.math.uchicago.edu/~may/PAPERS/MNApril20.pdf

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.