2006/12/12 by Pierre Guillot, Guillot, Pierre
Mathematics · #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.math/0612327
openalex publication_date 2006/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a collection of operations on the cohomotopy of any space, with which it becomes a "beta-ring", an algebraic structure analogous to a lambda-ring. In particular, this ring possesses Adams operations, represented by maps on the infinite loop space of the sphere spectrum. We compute their effect in homotopy on the image of J, and in mod 2 cohomology. The motivation comes from the interpretation of the symmetric group as the general linear group of the "field with one element", which leads to an analogy between cohomotopy and algebraic K-theory. A good deal of this article may be considered as a survey of the theory of beta-rings.