The concordance diffeomorphism group of real projective space

Abstract:

Let ${P_r}$ be r-dimensional real projective space with r odd, and let ${\pi _0}{\text {Diff}^ + }:{P_r}$ be the group of orientation preserving diffeomorphisms ${P_r} \to {P_r}$ factored by the normal subgroup of those concordant (= pseudoisotopic) to the identity. The main theorem of this paper is that for $r \equiv 11 \bmod 16$ the group ${\pi _0}{\text {Diff}^ + }:{P_r}$ is isomorphic to the homotopy group ${\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}})$, where $k = d{2^L} - r - 1$ with $L \geq \varphi ((r + 1)/2)$ and $d{2^L} \geq r + 1$. The function $\varphi$ is denned by $\varphi (l) = \{ i|0 < i \leq l,i \equiv 0,1,2,4 \bmod (8)\}$. The theorem is proved by introducing a cobordism version of the mapping torus construction; this mapping torus construction is a homomorphism $t:{\pi _0}{\text {Diff}^ + }:{P_r} \to {\Omega _{r + 1}}(v)$ for $r \equiv 11 \bmod 16$ and ${\Omega _{r + 1}}(v)$ a suitable Lashof cobordism group. It is shown that t is an isomorphism onto the torsion subgroup ${\Omega _{r + 1}}(v)$, and that this subgroup is isomorphic to ${\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}})$ as above. Then one reads off from Mahowald’s tables of ${\pi _{n + m}}({P_\infty }/{P_{m - 1}})$ that ${\pi _0}{\text {Diff}^ + }:{P_{11}} = {Z_2}$ and ${\pi _0}{\text {Diff}^ + }:{P_{27}} = 6{Z_2}$.