I am teaching a class on topology and want to introduce Steenrod Operations. I have talked about simplicial sets and classifying spaces of groups but have not talked about Eilenberg–MacLane spaces. What is the easiest/your favorite way of showing Steenrod operations exist? Is there a good way of constructing an explicit chain map $C_*(E S_2) \otimes_{\mathbb Z[S_2]} (C^*(X)\otimes C^*(X)) \to C^*(X)$ for some model of $E S_2$ that gives Steenrod operations?
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3$\begingroup$ Steenrod's original construction (with Z/2 coefficients) was in terms of "cup-i" products, which are explicit bilinear operations on $C^*(X)$ generalizing the construction of cup products. I don't know what the original references for this look like: you can also look at the papers of Berger-Fresse arxiv.org/abs/math/0109158 (esp. Theorem 2.1.1) or McClure-Smith arxiv.org/abs/math/0106024 (esp. 2.11) for a modern point of view, which in general gives complicated formulas, but simplifies reasonably well in the case you're interested in. $\endgroup$– Charles RezkCommented Oct 21, 2017 at 16:42
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2$\begingroup$ For what its's worth the cup-i products are constructed in Steenrod and Epstein "Cohomology Operations - Lectures by Steenrod" in chapter VII, pg 97. Mosher and Tangora follow this approach fairly closely in "Cohomology Operations and Applications to Homotopy Theory" chapter II (see pg 15 for your specific query). There is also May's approach in "A General Algebraic Approach to Steenrod Operations, and Smith's paper "An Algebraic Introduction to the Steenrod Algebra", although I can't really recall many specific details of the latter two. $\endgroup$– TyroneCommented Oct 21, 2017 at 16:58
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2$\begingroup$ I know a few main methods to do this. The first is to use explicit formulas for the cup-i products like Charles mentioned; this is explicit and quick to go through, but it takes some work to verify that the formulas are valid. A second is to use geometric constructions to produce an equivariant map $ES_2 \times K(Z/2,n)^2 \to K(Z/2,2n)$ and take chains; this is done in Hatcher's book, but it takes some more serious homotopy theory. $\endgroup$– Tyler LawsonCommented Oct 21, 2017 at 17:28
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7$\begingroup$ A third is to use something like the method of acyclic models, which Mosher and Tangora use. This is formula-free, and the method of acyclic models is a tool that can be used in a lot of other areas. This is usually my preferred method--the problems are that the proof seems like magic (it provides almost no intuition) and the method has fallen by the wayside in the past couple of decades since Hatcher's text became the standard. $\endgroup$– Tyler LawsonCommented Oct 21, 2017 at 17:34
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1$\begingroup$ I'm giving some talks on this right now to our in-house topology/geometry seminar. I actually went through all three formulations: the original cup_i products (without giving the explicit formulas), the dual approach using the map $W_*\otimes S_*\to S_*\otimes S_*$ as in Mosher-Tangora, and an Eilenberg-MacLane space version (though I was upfront about having no idea how to prove these are all equivalent). I felt that the Mosher-Tangora version is the one where you can show the most while invoking the least from "outside" - just the equivariant version of acyclic models. $\endgroup$– Greg FriedmanCommented Oct 21, 2017 at 23:07
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2 Answers
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I like to observe that the diagonal map $X\to X\times X$ is $\mathbb{Z}/2$-equivariant, hence induces a map of homotopy colimits. Analyzing this map with $X = K( \mathbb{Z}/2,n)$ gives the squares. The Adem relations follow from an extra symmetry (transpose) of the double composition $X \to X^4$.
answered Nov 2, 2017 at 18:18
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Its nice to look also at Bott's early paper "On symmetric products and the Steenrod squares. " Ann. of Math. (2) 57, (1953). 579–590.
He uses an early version of Smith theory. Depending on how you do Smith Theory the homotopy quotient construction lurking somewhere. Anyway worth look.
answered Nov 2, 2017 at 23:40