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.

Floyd's manifold is a conjugation space

“…To prove our main result is now, we will verify the homotopical criterion for being a conjugation space from [6]. Let HF denote the genuine C 2 -equivariant spectrum associated to the constant functor with values F 2 , the field of two elements.…”

“…As a C 2 -representation sphere S 3 = S 2σ+1 = S 2ρ−1 in the notation of [6], where σ stands for the sign representation and ρ for the regular representation. Notice that because of (⋆), the action of S 1 on S 3 restricts to an action of C 2 ∼ = {±1} ⊂ S 1 on (S 3 ) C 2 = S 1 , more precisely it restricts to the antipodal action on the fixed circle.…”

“…This yields an action of C 2 on F 10 . The second and final step consists then in verifying the "cohomological purity" in the genuine equivariant stable sense as explained in our joint work with Ricka, [6]. A key ingredient here is Clover May's splitting result, [5].…”

Realizing doubles: a conjugation zoo

“…This is trivially seen to be a conjugation space. From the equivariant stable perspective, [19], this is equally obvious since the sphere we just described is precisely the representation sphere S nρ where ρ is the regular representation, sum of the trivial and the sign representation. As it is now standard, if V is a finite dimensional real representation of C 2 , we denote by S V its one-point compactification.…”

“…We exhibit such exotic spaces, but notice that their exoticity vanishes when completed at 2. In fact, in view of our recent characterization of conjugation spaces, [19], a better way to understand this is to work in the stable C 2 -equivariant homotopy category. Smashing any conjugation space with HF, the genuine equivariant Eilenberg-Mac Lane spectrum for constant Z/2 coefficients, we get a spectrum that splits as a wedge of copies of suspensions Σ nρ HF where ρ is the regular representation; in this stable setting any conjugation space behaves as if it were built from conjugation cells.…”