The Kervaire manifold M_K is a certain closed simply-connected piecewise linear

manifold of dimension 4k+2 with the same homology as the product of a pair of (2k+1)-spheres.

Here are two questions about the Kervaire manifold:

Q1 When the does M_K admit a smooth structure?

Q2 When does M_K admit a free orientation preserving involution?

Q1 is a formulation of the Kervaire invariant problem which is now solved in all dimensions except 126 thanks to the recent breakthrough of Hill, Hopkins and Ravenel.

In this talk, I will report on joint research with Ian Hambleton on Q2 and its relationship with Q1. For

example, if Q1 has a positive answer then so does Q2. One of our main results is that Q2 has a positive answer in dimension 126.

In a new development, we conjecture that if Q2 has a positive answer in dimension 2^{k+1}-2, then

Q1 has a positive answer in dimension 2^k - 2.

Our work is based on deep theorems in surgery of Brumfiel, Madsen and Milgram from the 1970s.