The concordance diffeomorphism group of real projective space

Abstract

Let P r {P_r} be r-dimensional real projective space with r odd, and let π 0 Diff + : P r {\pi _0}{\text {Diff}^ + }:{P_r} be the group of orientation preserving diffeomorphisms P r → P r {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 ≡ 11 mod 16 r \equiv 11 \bmod 16 the group π 0 Diff + : P r {\pi _0}{\text {Diff}^ + }:{P_r} is isomorphic to the homotopy group π r + 1 + k ( P ∞ / P k − 1 ) {\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}}) , where k = d 2 L − r − 1 k = d{2^L} - r - 1 with L ≥ φ ( ( r + 1 ) / 2 ) L \geq \varphi ((r + 1)/2) and d 2 L ≥ r + 1 d{2^L} \geq r + 1 . The function φ \varphi is denned by φ ( l ) = { i | 0 > i ≤ l , i ≡ 0 , 1 , 2 , 4 mod ( 8 ) } \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 : π 0 Diff + : P r → Ω r + 1 ( v ) t:{\pi _0}{\text {Diff}^ + }:{P_r} \to {\Omega _{r + 1}}(v) for r ≡ 11 mod 16 r \equiv 11 \bmod 16 and Ω r + 1 ( v ) {\Omega _{r + 1}}(v) a suitable Lashof cobordism group. It is shown that t is an isomorphism onto the torsion subgroup Ω r + 1 ( v ) {\Omega _{r + 1}}(v) , and that this subgroup is isomorphic to π r + 1 + k ( P ∞ / P k − 1 ) {\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}}) as above. Then one reads off from Mahowald’s tables of π n + m ( P ∞ / P m − 1 ) {\pi _{n + m}}({P_\infty }/{P_{m - 1}}) that π 0 Diff + : P 11 = Z 2 {\pi _0}{\text {Diff}^ + }:{P_{11}} = {Z_2} and π 0 Diff + : P 27 = 6 Z 2 {\pi _0}{\text {Diff}^ + }:{P_{27}} = 6{Z_2} .