Conjugation Spaces are Cohomologically Pure

Abstract

Conjugation spaces are equipped with an involution such that the fixed points have the same mod 2 cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by 2, generalizing the classical examples of complex projective spaces under complex conjugation. Using tools from stable equivariant homotopy theory we provide a characterization of conjugation spaces in terms of purity. This conceptual viewpoint, compared to the more computational original definition, allows us to recover all known structural properties of conjugation spaces.

arXiv comment39 pages. This version corrected some misprints and a few misleading proofs

References in corpus (11)

• Global homotopy theory
• Grothendieck-Neeman duality and the Wirthmüller isomorphism
• Conjugation spaces
• Equivariant Universal Coefficient and Kunneth Spectral Sequences
• Steenrod squares on conjugation spaces
• A structure theorem for $RO(C_2)$-graded Bredon cohomology
• Derived induction and restriction theory
• $C_2$-equivariant Homology Operations: Results and Formulas
• Motivic modular forms from equivariant stable homotopy theory
• Realizing doubles: a conjugation zoo
• The $RO(C_2)$-graded cohomology of $C_2$-surfaces in $\underline{\mathbb{Z}/2}$-coefficients

Cited by in corpus (2)

• Odd primary analogs of Real orientations
• Realizing doubles: a conjugation zoo