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Some examples of vanishing Yamabe invariant and minimal volume, and collapsing of inequivalent smoothings and PL-structures

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Abstract

In this short note, exploits of constructions of \(\mathcal {F}\)-structures coupled with technology developed by Cheeger–Gromov and Paternain–Petean are seen to yield a procedure to compute minimal entropy, minimal volume, Yamabe invariant and to study collapsing with bounded sectional curvature on inequivalent smooth structures and inequivalent PL-structures within a fixed homeomorphism class. We compute these fundamental Riemannian invariants for every high-dimensional smooth manifold on the homeomorphism class of any smooth manifold that admits a Riemannian metric of zero sectional curvature. This includes all exotic and all fake tori of dimension greater than four. We observe that the minimal volume is not an invariant of the smooth structures, yet the Yamabe invariant does discern the standard smooth structure from all the others. We also observe that the fundamental group places no restriction on the vanishing of the minimal volume and collapse with bounded sectional curvature for high-dimensional manifolds.

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Acknowledgments

We are indebted to Jim Davis for kindly bringing [13, Appendix] to our attention, without which we could have not proven our main result as stated. We thank Frank Quinn, Andrew Ranicki, and Terry Wall for helpful e-mail correspondence. We thank Bernd Ammann, Oliver Baues, Fernando Galaz-García, Marco Radeschi, and Wilderlich Tuschmann for interesting conversations. We gratefully acknowledge support from the University of Fribourg and the organizers of its Riemannian Topology Seminar 2015 for a very pleasant and productive meeting during which part of this paper was written.

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  1. Scuola Internazionale Superiori di Studi Avanzati (SISSA), Via Bonomea 265, 34136, Trieste, Italy

    Rafael Torres

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  1. Rafael Torres
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Torres, R. Some examples of vanishing Yamabe invariant and minimal volume, and collapsing of inequivalent smoothings and PL-structures. Ann Glob Anal Geom 49, 177–194 (2016). https://doi.org/10.1007/s10455-015-9486-9

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