An almost-complex manifold
supports an involution if there is a differentiable self-map on the manifold of period
two. The differential of the map acts on the coset space of the almost-complex
structures on M by inner automorphism. This action is also of period two. If
the almost-complex structure is sent to its conjugate, the manifold with
structure, together with the given involution is called a conjugation. Any linear
involution of Euclidean space may be used to stabilize this situation, giving
a cobordism theory of exotic conjugations. The question considered here
is: What is the image in complex cobordism of the functor which forgets
equivariance. The result shown in the next section is: If a stably almost-complex
manifold supports an exotic conjugation, every characteristic number is
even.
