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Where in literature can one find a construction of Steenrod reduced powers (for an odd $p$) that

(1) works for the singular cohomology of arbitrary topological spaces (or, more generally, for the cohomology of simplicial sets) and

(2) does not use homotopy-theoretic constructions (such as $K(\pi,n)$ spaces)?

I like the approach of the Steenrod--Epstein book. Unfortunately, the main construction is done there for the cellular cohomology of finite regular cell complexes. Then the operations are extended to infinite regular complexes and then to arbitrary CW-complexes (using approximation by a simplicial complex); then this is applied to the singular simplicial set of a topological space. This way seems somewhat indirect if we are interested in singular cohomology.

Another approach is given in Switzer's "Algebraic topology" for $p=2$. It starts with the words "By the acyclic model theorem, there exists a natural equivariant chain map $W\otimes S(X)\to S(X)\otimes S(X)$" (where $S(X)$ is the singular chain complex and $W$ is the free resolution of the group $\mathbb Z/2$). Some other textbooks also use this approach. Unfortunately, I could not find an exposition for $p$ odd.

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  • $\begingroup$ May's "general algebraic approach to Steenrod operations" is an answer, surely. Possibly there is a simpler exposition aimed at topological spaces. $\endgroup$
    – Semen Podkorytov
    May 19, 2013 at 12:13
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    $\begingroup$ You might like to look at "Multivariable cochain operations and little n-cubes" by McClure and Smith. It is written for people who know a lot already but I think it may be possible to adapt it to give a more concrete and elementary approach to the Steenrod operations. I think that there also exists an explicit approach for $p>2$ written in the 1960s by some Japanese authors, but I looked at it briefly a long time ago and found it completely indigestible. $\endgroup$
    – Neil Strickland
    Jun 5, 2013 at 19:06

3 Answers 3

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P. Selick, Introduction to Homotopy Theory, Chapter 14.

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Useful references

A. Dold, Über die Steenrodschen Kohomologieoperationen, Ann. Math. (1961), Russian translation: Matematika 7:6 (1963)

S. Gitler, Cohomology operations with local coefficients, Am. J. Math. (1963)

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Lurie's notes on the Sullivan conjecture: http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture2.pdf constructs the total operations and then proves the properties in later lectures.

One does not need to know what an $\mathbb{F}_2$-module spectrum is and just think of them as cochain complexes (as he explains)

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  • $\begingroup$ I am particularly interested in the case of odd p. The lecture you point to is about p=2. $\endgroup$
    – Semen Podkorytov
    Oct 12, 2014 at 17:35

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