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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The concordance diffeomorphism group of real projective space

Author: Robert Wells
Journal: Trans. Amer. Math. Soc. 192 (1974), 319-337
MSC: Primary 57D50
MathSciNet review: 0339224
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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\vert < 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}$.

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Keywords: Diffeomorphism, concordance, pseudoisotopy, Lashof cobordism groups, Moore-Postnikov decomposition, mapping torus, surgery, Wall groups
Article copyright: © Copyright 1974 American Mathematical Society