Circle-valued Morse theory for complex hyperplane arrangements

Toshitake Kohno; Andrei Pajitnov

doi:10.1515/forum-2013-0032

Published by De Gruyter July 13, 2013

Circle-valued Morse theory for complex hyperplane arrangements

Toshitake Kohno and Andrei Pajitnov

From the journal Forum Mathematicum

https://doi.org/10.1515/forum-2013-0032

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Abstract

Let 𝒜 be an essential complex hyperplane arrangement in ℂn, and H denote the union of the hyperplanes. We develop the real-valued and circle-valued Morse theory on the space M=ℂn∖H and prove, in particular, that M has the homotopy type of a space obtained from a manifold V fibered over a circle, by attaching |χ(M)| cells of dimension n. We compute the Novikov homology H^*(M,ξ) for a large class of homomorphisms ξ:π1(M)→ℝ.

Keywords: Arrangement; Novikov homology; Morse function

MSC: 14N20; 32S22; 57R45; 57R19

Funding source: World Premier International Research Center Initiative (WPI Program), MEXT, Japan

The work was supported by World Premier International Research Center Initiative (WPI Program), MEXT, Japan. The work was accomplished during the second author's stay at the Graduate School of Mathematical Sciences, the University of Tokyo in the autumn of 2010 and 2011. The second author is grateful to the Graduate School of Mathematical Sciences for warm hospitality. The authors thank A. Suciu for valuable discussions as well as the anonymous reviewer for helpful comments on the manuscript.

Received: 2013-3-3

Revised: 2013-5-27

Published Online: 2013-7-13

Published in Print: 2015-7-1

Circle-valued Morse theory for complex hyperplane arrangements

Abstract

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