Vol. 53, No. 2, 1974

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Conjugations on stably almost complex manifolds

Allan L. Edelson

Vol. 53 (1974), No. 2, 373–377

A stably almost complex structure on a smooth manifold M is an automorphism J : τM 𝜃k τM 𝜃k for some k 0, covering the identity map on M, and satisfying J2 = l. If k = 0,J is an almost complex structure. An involution T : M M is a conjugation of (M,J) if there exists an involution α;𝜃k 𝜃k covering T, such that Tα is conjugate linear, i.e., (Tα) J = J (Tα). The bordism theory of conjugations has been studied by R. Stong. In §2 of this article it is shown that every closed n-manifold can be realized as the fixed point set of a conjugation on a closed, 2n-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 2n-manifold is n-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.

Mathematical Subject Classification
Primary: 57D85
Received: 6 April 1973
Revised: 7 August 1973
Published: 1 August 1974
Allan L. Edelson