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April 2016 Inertia groups and smooth structures of ($n-1$)-connected $2n$-manifolds
Kasilingam Ramesh
Osaka J. Math. 53(2): 309-319 (April 2016).


Let $M^{2n}$ denote a closed ($n-1$)-connected smoothable topological $2n$-manifold. We show that the group $\mathcal{C}(M^{2n})$ of concordance classes of smoothings of $M^{2n}$ is isomorphic to the group of smooth homotopy spheres $\bar{\Theta}_{2n}$ for $n=4$ or $5$, the concordance inertia group $I_{c}(M^{2n})=0$ for $n=3$, $4$, $5$ or $11$ and the homotopy inertia group $I_{h}(M^{2n})=0$ for $n=4$. On the way, following Wall's approach [16] we present a new proof of the main result in [9], namely, for $n=4$, $8$ and $H^{n}(M^{2n};\mathbb{Z})\cong \mathbb{Z}$, the inertia group $I(M^{2n})\cong \mathbb{Z}_{2}$. We also show that, up to orientation-preserving diffeomorphism, $M^{8}$ has at most two distinct smooth structures; $M^{10}$ has exactly six distinct smooth structures and then show that if $M^{14}$ is a $\pi$-manifold, $M^{14}$ has exactly two distinct smooth structures.


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Kasilingam Ramesh. "Inertia groups and smooth structures of ($n-1$)-connected $2n$-manifolds." Osaka J. Math. 53 (2) 309 - 319, April 2016.


Published: April 2016
First available in Project Euclid: 27 April 2016

zbMATH: 1348.57045
MathSciNet: MR3492800

Primary: 57R50 , 57R55 , 57R60 , 57R65

Rights: Copyright © 2016 Osaka University and Osaka City University, Departments of Mathematics

Vol.53 • No. 2 • April 2016
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