Conjugations on 6-manifolds

Abstract

Conjugation spaces are spaces with an involution such that the fixed point set of the involution has Z 2 -cohomology ring isomorphic to the Z 2 -cohomology of the space itself, with the difference that all degrees are divided by two (e.g. CP n with the complex conjugation has RP n as fixed point set). One also requires that a certain conjugation equation is fulfilled. We give a new characterisation of conjugation spaces and apply it to the following realization problem: given M, a closed orientable 3-manifold, does there exist a simply connected 6-manifold X and a conjugation on X with fixed point set M? We give an affirmative answer. Mathematics Subject Classification (2000)57R91 · 55M35 · 55N91

Conjugations on 6-manifolds with free integral cohomology

“…We deduce that H 2 (X) ∼ = Coker(H 2 (S 2 ) → H 2 (V 0 )). But in [Olb07] we have seen that if W 0 has normal 2-type B, then this cokernel is isomorphic to Z m−1 − . So we have constructed a conjugation on a simply connected spin manifold X with fixed point set M .…”

“…In particular we see that the normal 2-type of W 0 is equal to the normal 2-type of W . By the Mayer-Vietoris sequence for X = M × D 3 ∪ V 0 , we see that H 2 (X) is a free Z-module on which the conjugation acts by multiplication with −1 (see [Olb07]). Since H 1 (X) and H 2 (X) are free over Z, Poincaré duality implies that all homology of X is free over Z.…”

“…So we have to look at the double cover of the map M × RP 2 → Q m . Going through our construction of this map [Olb07], we see that the map M × S 2 → (CP ∞ ) m maps the 2-skeleton of M to a point, and S 2 = CP 1 diagonally into (CP ∞ ) m . Hence the map…”

“…We can compute the transfer map using the Atiyah-Hirzebruch spectral sequence. For the facts needed and not proved here see Zubr [Zub75] (for the computation of ΩB 6 ) and [Olb07] (for the computation of Ω B 6 ).…”

“…The problem of realizing 3-manifolds as fixed point sets of conjugations was considered in [Olb07]. The present paper focuses on the identification of the resulting conjugation space.…”