Involutions fixing many components: a small codimension phenomenon

Abstract

Let $$T{:}M \rightarrow M$$ be a smooth involution on a closed smooth m-dimensional manifold and $$F = \bigsqcup _{j=0}^n F^j$$ the fixed point set of T, where $$F^j$$ denotes the union of those components of F having dimension j, and thus, $$n < m$$ is the dimension of the component of F of largest dimension. Denote by $$\pi _0(F)$$ the set of dimensions occurring in F. If $$j \in \pi _0(F)$$ , $$0 \le j \le n$$ , we assume that the normal bundle of $$F^j$$ in M does not bound, because otherwise, $$F^j$$ can be removed via an equivariant surgery. In this paper, we prove the following results, which characterize small codimension phenomena: suppose that F has one of the following forms: (i) $$0 \in \pi _0(F)$$ , and all the other components of F (including the top-dimensional, with dimension n) are odd-dimensional; (ii) $$1 \in \pi _0(F)$$ , and all the other components of F (including the top-dimensional) are even-dimensional. If k denotes the codimension of $$F^n$$ , then $$k\le 1$$ in the first case, and $$k\le 2$$ in the second case. Furthermore, very simple examples will show that these results are best possible. Other results concerning the small codimension phenomenon are found in the literature, where F has two, three, or four components. Together with our results, they inspire a general conjecture that will be presented in Sect. 1, and if valid, this conjecture is best possible. In fact, (i) and (ii) are the cases $$j=0$$ and $$j=1$$ of the conjecture in question. Unlike the above-mentioned literature results, note that in our case, the number of components of F may not be limited as a function of n. In fact, in the first case, if S is any subset of the set $$\{i\ | \ 0<i <n, \ i \ \text {odd} \}$$ , then we allow that $$S \subset \pi _0(F)$$ . In the second case, if S is any subset of the set $$\{i\ | \ 0 \le i <n, \ i \ \text {even} \}$$ , we allow that $$S \subset \pi _0(F)$$ . Because of this, to show that the results are not vacuous and that there many involutions to which the theorem applies, we will show that there are examples where all possible dimensions occur, that is, with S maximal.