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The Hirzebruch signature theorem for conical metrics. (English) Zbl 1376.57028

Ballmann, Werner (ed.) et al., Arbeitstagung Bonn 2013. In memory of Friedrich Hirzebruch. Proceedings of the meeting, Bonn, Germany, May, 22–28, 2013. Basel: Birkhäuser/Springer (ISBN 978-3-319-43646-3/hbk; 978-3-319-43648-7/ebook). Progress in Mathematics 319, 1-15 (2016).
The various signature theorems in topology relate the Euler characteristic to formulae using the characteristic classes of a manifold. In the presence of singularities the formulae will fail to hold although each side of the equation may be defined. The difference between the two sides is then called the “defect”. There are three noteworthy cases when the defect is well-defined. This paper considers the case where the space is a manifold but the characteristic classes are computed using a Riemannian metric with singularities to compute the differential forms representing the characteristic classes. Specifically, in this paper the singularities are conical. The main result computes the defect in this case.
For the entire collection see [Zbl 1364.00032].

MSC:

57R19 Algebraic topology on manifolds and differential topology
57R20 Characteristic classes and numbers in differential topology
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
57-03 History of manifolds and cell complexes
01A60 History of mathematics in the 20th century
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