2005/09/21 by Joost van Hamel, van Hamel, Joost · 1 citation
Mathematics · #55M35 #55N91 #57R91 #57S17 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55M35 #msc:55N91 #msc:57R91 #msc:57S17
paper · pdf · doi:10.48550/arxiv.math/0509498
arxiv created 2005/09/21 · openalex publication_date 2005/09/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For certain manifolds with an involution the mod 2 cohomology ring of the set of fixed points is isomorphic to the cohomology ring of the manifold, up to dividing the degrees by two. Examples include complex projective spaces and Grassmannians with the standard antiholomorphic involution (with real projective spaces and Grassmannians as fixed point sets). Hausmann, Holm and Puppe have put this observation in the framework of equivariant cohomology, and come up with the concept of "conjugation spaces", where the ring homomorphisms arise naturally from the existence of what they call "cohomology frames". Much earlier, Borel and Haefliger had studied the degree-halving isomorphism between the cohomology rings of complex and real projective spaces and Grassmannians using the theory of complex and real analytic cycles and cycle maps into cohomology. The main result in the present note gives a (purely topological) connection between these two results and provides a geometric intuition between the concept of a cohomology frame. In particular, we see that if every cohomology class on a manifold X with involution is the Thom class of an equivariant topological cycle of codimension twice the codimension of its fixed points (inside the fixed point set of X), these topological cycles will give rise to a cohomology frame.