Hirzebruch signature theorem and exotic smooth structures on the 7-sphere

Authors

  • Guifré Sánchez Serra Universitat Autònoma de Barcelona

Keywords:

characteristic classes, manifold cobordism, exotic spheres.

Abstract

The existence of non-standard smooth structures on Sn was not proven until 1956, when J. Milnor presented an explicit construction for the case n = 7, [4]. Until then, it was assumed that there was no fundamental difference between topological and smooth spheres. This had profound implications in the field of manifold and algebraic topology, and was immediately endorsed by subsequent research, which lead to the characterization of the so called groups of homotopy spheres, [2]. One of the results that made Milnor's approach possible was Hirzebruch's signature theorem, which gives a formula to compute the signature of a (smooth) compact oriented manifold. The aim of this work is to contextualize this theorem, as well as to show its role in the construction of the first exotic 7-spheres.

Keywords: characteristic classes, manifold cobordism, exotic spheres.

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Author Biography

Guifré Sánchez Serra, Universitat Autònoma de Barcelona



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How to Cite

Sánchez Serra, G. (2021). Hirzebruch signature theorem and exotic smooth structures on the 7-sphere. Reports@SCM, 6(1), 11–22. Retrieved from https://revistes.iec.cat/index.php/reports/article/view/149332

Issue

Vol. 6 No. 1 (2021)

Section

Articles

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