21 - When is P_{n, k} neutral?

Summary

This section is based on joint work with Sutherland. Recall that d_n denotes the self-map of P_n,k defined by reflection in the last coordinate hyperplane. We say that P_{n, k} is neutral (elsewhere outsimple) if d_n ≃ 1, and define S-neutral similarly. If n and k are both odd then P_n,k is neutral, as remarked in §7. If n is even then d_n has degree −1 on the integral homology H_n−1,(P_n,k) = Z, and so P_n,k is not S-neutral. Thus the interest resides in the case when n is odd and k even. Notice that P_k+1,k = P^k is neutral for all even values of k. In the course of §6 we have already proved

Proposition (21.1). Suppose that P_n,k is S-neutral, where n is odd and k even. Then P_m+n,kand P_m+k−n,kare S-neutral, whenever m ≡ 0 mod â_k.

Here â_k is as in (1.10). Now consider V_n,k as a Z₂ -space under the outer automorphism which changes the sign of the last row and column of each matrix. Since the inclusion P_n,k → V_n,k is a Z₂ - map it follows at once from (3. 4) that P_n,k is neutral if V_n,k is neutral and n ≥ 2k.