Pontryagin cohomology operation

In mathematics, a Pontryagin cohomology operation is a cohomology operation taking cohomology classes in H²ⁿ(X,Z/p^rZ) to H^2pn(X,Z/p^r+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. Oxford. 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), "Pontryagin square", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 
• Pontryagin, L. (1942), "Mappings of the three-dimensional sphere into an n-dimensional complex", C. R. (Doklady) Acad. Sci. URSS (N. S.), 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: 266–269, doi:10.1073/pnas.42.5.266, ISSN 0027-8424, JSTOR 89856, MR 0079254 
• Whitehead, J. H. C. (1949), "On simply connected, 4-dimensional polyhedra", Commentarii Mathematici Helvetici, 22: 48–92, doi:10.5169/seals-19190, ISSN 0010-2571, MR 0029171