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 T∗⊕α is conjugate linear,
i.e., (T∗⊕ α) ∘ J = −J ∘ (T∗⊕ α). 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.
