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Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.52 no.2 Bogotá Jul./Dec. 2018
https://doi.org/10.15446/recolma.v52n2.77161
Original articles
A proof of the Adem relations
Una demostración de las relaciones de Adem
1 Watson Research Center, Computational Genomics, IBM Thomas J., Yorktown Heights, NY, USA. e-mail: aldo.guzman.saenz@ibm.com
2 Centro de Investigación y de Estudios Avanzados del IPN, Departamento de Matemáticas. Mexico City, 07360. Mexico. e-mail: xico@math.cinvestav.mx
We give an alternative proof of the Bullett-Macdonald identity for the Steenrod squares, which is in turn equivalent to the Adem relations. The main idea is to show that the iterated total squaring operation S 2: H n (X) → H 4n(X × BZ2 × BZ2) is the restriction of a total fourth-power operation T: H n(X) → H 4n(X × BΣ4).
Keywords: Adem relations; total squaring operation
Damos una demostración alternativa de la identidad de Bullet-Macdonald para los cuadrados de Steenrod, la que a su vez es equivalente a las relaciones de Adem. La idea principal es mostrar que la iteración de la operación cuadrado total S2: H n(X) → H 4n(X × BZ2 × BZ2) es la restricción de una operación cuadrado total T: H n(X) → H 4n(X × BΣ4).
Palabras clave: Relaciones de Adem; operación cuadrado total
Referencias
[1] A. Adem and R. J. Milgram, Cohomology of finite groups, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 309, Springer-Verlag, Berlin, 2004. [ Links ]
[2] J. Adem, The iteration of the Steenrod squares in algebraic topology, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 720-726. [ Links ]
[3] S. R. Bullett and I. G. Macdonald, On the Adem relations, Topology 21 (1982), no. 3, 329-332. [ Links ]
[4] Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. (1968), no. 34, 105-112. [ Links ]
[5] N. E. Steenrod, Cohomology operations, Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. [ Links ]
Received: August 14, 2018; Accepted: September 16, 2018
This is an open-access article distributed under the terms of the Creative Commons Attribution License
