In mathematics, a Pontryagin cohomology operation is a cohomology operation taking cohomology classes in H2n(X,Z/prZ) to H2pn(X,Z/pr+1Z) for some prime number p. When p=2 these operations were introduced by Pontryagin (1942) and were named Pontrjagin squares by Whitehead (1949) (with the term "Pontryagin square" also being used). They were generalized to arbitrary primes by Thomas (1956).
See also Steenrod operation
References Browder, William; Thomas, E. (1962), "Axioms for the generalized Pontryagin cohomology operations", The Quarterly Journal of Mathematics, Second Series, 13 (1): 55–60, doi:10.1093/qmath/13.1.55, ISSN 0033-5606, MR 0140103 Malygin, S.N.; Postnikov, M.M. (2001) [1994], "Pontryagin square", Encyclopedia of Mathematics, EMS Press Pontryagin, L. (1942), "Mappings of the three-dimensional sphere into an n-dimensional complex", C. R. (Doklady) Acad. Sci. URSS, New Series, 34: 35–37, MR 0008135 Thomas, Emery (1956), "A generalization of the Pontrjagin square cohomology operation", Proceedings of the National Academy of Sciences of the United States of America, 42 (5): 266–269, Bibcode:1956PNAS...42..266T, doi:10.1073/pnas.42.5.266, ISSN 0027-8424, JSTOR 89856, MR 0079254, PMC 528270, PMID 16589865 Whitehead, J. H. C. (1949), "On simply connected, 4-dimensional polyhedra", Commentarii Mathematici Helvetici, 22: 48–92, doi:10.1007/bf02568048, ISSN 0010-2571, MR 0029171, S2CID 121723000