Celebratio Mathematica

phdthesis E. Thomas: A generalization of the Pontrjagin square cohomology operation. Ph.D. thesis, Princeton University, 1955. Advised by N. E. Steenrod. Published (under a different title) as Memoirs of the American Mathematical Society 27 (1957). See also Proc. Nat. Acad. Sci. U.S.A. 42:5 (1956).

N. E. Steenrod and E. Thomas: “Cohomology operations derived from cyclic groups,” Comment. Math. Helv. 32 (1957), pp. 129–152. MR 0092148 Zbl 0090.39101 article

In a previous paper [1957], Steenrod defined a family of cohomology operations, called reduced powers, each being associated with some permutation group. It was also shown that these operations have a basis, in the sense of composition, consisting of, firstly, four primitive types of operations (which are: addition, cup-product, homomorphisms induced by coefficient homomorphisms, and Bockstein–Whitney coboundary operators) and, secondly, those reduced powers associated with cyclic permutation groups having degree \( p \) and order \( p \) where \( p \) ranges over primes.

In this paper, we shall improve the result by showing that there is a smaller basis consisting of the same primitive operations and only particular operations associated with cyclic groups: namely, for each prime \( p \) the cyclic reduced powers \[ \mathcal{P}^i: H^q(K;Z_p) \to H^{q+2i(p-1)}(K;Z_p),\qquad i = 0, 1, \dots , \] and the Pontrjagin \( p \)-th powers \[ \mathfrak{P}_p: H^{2q}(K;Z_{p^k}) \to H^{2pq}(K; Z_{p^{k+1}}). \] The latter were defined for \( p = 2 \) by Pontrjagin [1942], and generalized for \( p > 2 \) by Thomas [1956]. When \( p = 2 \), \( \mathcal{P}^i \) is usually written \( \operatorname{Sq}^{2i} \).

E. Thomas: “The Thom–Massey approach to embeddings,” pp. 283–307 in The Steenrod algebra and its applications (Battelle Memorial Institute, Columbus, OH, 30 March–4 April 1970). Edited by F. P. Peterson. Lecture Notes in Mathematics 168. Springer (Berlin), 1970. Proceedings of a conference to celebrate N. E. Steenrod’s sixtieth birthday. MR 0287560 Zbl 0216.45302 incollection

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