THE CONCORDANCE DIFFEOMORPHISM GROUP OF REAL PROJECnVE SPACE

Let t, be r-dimensional real projective space with r odd, and let irnDiff* : Pr be the group of orientation preserving difteomorphisms P,-* P, factored by the normal subgroup of those concordant (= pseudoiso topic) to the identity. The main theorem of this paper is that fores 11 mod 16 the group T0Diff+: P, is isomorphic to the homotopy group irr+i+k(&/Pk-\), where k dlL r 1 with L > r + 1. The function „ = Z2 and ir0Diff+: P„ = 6Z2. Let P. be r-dimensional real projective space, and let ir0 Diff+ : Pr be the group of orientation preserving diffeomorphisms Pr-* P, factored by the normal subgroup of those concordant (= pseudoisotopic) to the identity. The object of this paper is to determine an isomorphism from the group w0Diff: R for r = 11 mod 16 to the torsion subgroup of a certain Lashof cobordism group. To define the cobordism group, let the fibration v: P[l] -* BSO be such that P„ -* P[l] -** BSO is the /th stage in the Moore-Postnikov decomposition of the map Px -* BSO classifying an orientation of kyn, where yr -* Pr is the canonical line bundle, / = (r + l)/2 and k = d2 r 1 for L > r + 1; recall that <p(/) = # {/1 0 < i < l,i = 0,1,2,4 mod 8}. Let Qr+X(v) be the (r + l)st Lashof cobordism group of the composed fibration P[l] -** BSO -* BO as in [3] or [6]. Finally, let f : Pr -» P. be the diffeomorphism defined by the matrix 1 -1 Then the theorem is the following. Received by the editors September 7,1972 and, in revised form, January 15, 1973. AMS (MOS) subject classifications (1970). Primary 57D50, 57D90; Secondary 55E10.