Conjugations on stably almost complex manifolds

Abstract

A stably almost complex structure on a smooth manifold M is an automorphism J: τ M 0 θ k -> τ M 0 θ k for some k ^ 0, covering the identity map on ilf, and satisfying J 2 = -1.If A; = 0, J is an almost complex structure.An involution T:M->M is a conjugation of (M, J) if there exists an involution a: θ k -» θ* covering T, such that T*@a is conjugate linear, i.e., (T* 0α)oj = -Jo(T* ©α).The bordism theory of conjugations has been studied by R. Stong.In § 2 of this article it is shown that every closed %-manifold can be realized as the fixed point set of a conjugation on a closed, 2%-dimensional stably almost complex manifold.This should be compared to the result of Conner and Floyd that the fixed point set of a conjugation on an almost complex 2nm anifold is ^-dimensional, which is false for stably almost complex manifolds.The proof will use the following result:LEMMA 1.Every closed manifold is cobordant to the fixed point set of a conjugation on a closed, almost complex manifold.LEMMA 4.There is a stably almost complex submanifold VaB, invariant under the conjugation, with bV = VΓ\bB -S.Proof.The involution T on B is free, and S is a characteristic submanifold for the restriction T\ bB .There is a map f:bB/T~> P N ,