Symmetries of homotopy complex projective three spaces

Abstract:

We study symmetry properties of six-dimensional, smooth, closed manifolds which are homotopy equivalent to ${\mathbf {C}}{P^3}$. There are infinitely differentiably distinct such manifolds. It is known that if $m$ is an odd prime, infinitely many homotopy ${\mathbf {C}}{P^3}$’s admit ${{\mathbf {Z}}_m}$-actions whereas only the standard ${\mathbf {C}}{P^3}$ admits an action of the group ${{\mathbf {Z}}_m} \times {{\mathbf {Z}}_m} \times {{\mathbf {Z}}_m}$. We study the intermediate case of ${{\mathbf {Z}}_m} \times {{\mathbf {Z}}_m}$-actions and show that infinitely many homotopy ${\mathbf {C}}{P^3}$’s do admit ${{\mathbf {Z}}_m} \times {{\mathbf {Z}}_m}$-actions for a fixed prime $m$. The major tool involved is equivariant surgery theory. Using a transversality argument, we construct normal maps for which the relevant surgery obstructions vanish allowing the construction of ${{\mathbf {Z}}_m} \times {{\mathbf {Z}}_m}$-actions on homotopy ${\mathbf {C}}{P^3}$’s which are ${{\mathbf {Z}}_m} \times {{\mathbf {Z}}_m}$-homotopy equivalent to a specially chosen linear action on ${\mathbf {C}}{P^3}$. A key idea is to exploit an extra bit of symmetry which is built into our set-up in a way that forces the signature obstruction to vanish. By varying the parameters of our construction and calculating Pontryagin classes, we may construct actions on infinitely many differentiably distinct homotopy ${\mathbf {C}}{P^3}$’s as claimed.