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On characteristic classes of exotic manifold bundles. (English) Zbl 1458.55011

Given a smooth closed oriented manifold \(M\) and an exotic sphere \(\Sigma\) of the same dimension, one can form the connected sum \(M \# \Sigma\). Sometimes this yields a manifold diffeomorphic to \(M\), if \(\Sigma\) lies in the inertia group of \(M\), but often \(M \# \Sigma\) is not diffeomorphic to \(M\). In the latter case, it is an interesting question to what extent the cohomology \(H^*(B\mathrm{Diff}^+(M);\mathbb{Z})\) of the classifying space of the topological group of orientation-preserving diffeomorphisms of \(M\) differs from \(H^*(B\mathrm{Diff}^+(M \# \Sigma);\mathbb{Z})\).
In this paper, it is proven when \(M\) is simply-connected and of even dimension \(2n \geq 6\), then in a range these cohomology groups are isomorphic after replacing the coefficients \(\mathbb{Z}\) by \(\mathbb{Z}[\tfrac{1}{k}]\) with \(k\) the order of \(\Sigma\) in the finite group \(\Theta_{2n}\) of oriented exotic \(2n\)-spheres. This range depends on the “stable genus” of \(M\), in the sense of S. Galatius and O. Randal-Williams [J. Am. Math. Soc. 31, No. 1, 215–264 (2018; Zbl 1395.57044); Ann. Math. (2) 186, No. 1, 127–204 (2017; Zbl 1412.57026); Acta Math. 212, No. 2, 257–377 (2014; Zbl 1377.55012)]. Furthermore, infinitely many families of examples are constructed to illustrate that it is necessary to invert the order of \(\Sigma\).
The main input is the aforementioned work of Galatius and Randal-Williams. This distinguishes it from earlier work of W. G. Dwyer and R. H. Szczarba [Ill. J. Math. 27, 578–596 (1983; Zbl 0507.57016)], which uses smoothing theory instead to prove a similar result about the classifying space of the identity component of \(\mathrm{Diff}^+(M)\).

MSC:

55R40 Homology of classifying spaces and characteristic classes in algebraic topology
57R60 Homotopy spheres, Poincaré conjecture
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
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