Dimensions of fixed point sets of involutions

Abstract.

Suppose the fixed point set F of a smooth involution T:M → M on a smooth, closed and connected manifold M decomposes into two components Fⁿ and F² of dimensions n and 2, respectively, with n > 2 odd. We show that the codimension k of Fⁿ is small if the normal bundle of F² does not bound; specifically, we show that k≦ 3 in this case. In the more general situation where F is not a boundary, n (not necessarily odd) is the dimension of a component of F of maximal dimension and k is the codimension of this component, and fixed components of all dimensions j, 0≦ j≦ n, may occur, a theorem of Boardman gives that $k \leqq \frac{3}{2}n$.

In addition, we show that this bound can be improved to k≦ 1 (hence k = 1) for some specific values of n and some fixed stable cobordism classes of the normal bundle of F² in M; further, we determine in these cases the equivariant cobordism class of (M, T).