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Bruhat Numbers of a Strong Morse Function

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Abstract

Let f be a Morse function on a manifold M such that all its critical values are pairwise distinct. Given such a function (together with a certain choice of orientations) and a field \(\mathbb{F}\), we construct a set of nonzero elements of the field, which are called Bruhat numbers. Under certain acyclicity conditions on M, the alternating product of all the Bruhat numbers does not depend on f (up to sign); thus, it is an invariant of the manifold. For any typical one-parameter family of functions on M, we provide a relation that links the Bruhat numbers of the boundary functions of the family with the number of bifurcations happening along a path in the family. This relation generalizes the result from [1].

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REFERENCES

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ACKNOWLEDGMENTS

We are grateful to V. Vassiliev for careful reading of the manuscript and a number of useful comments that helped make the exposition more clear.

Funding

Temkin’s study was performed within the basic research program of the National Research University Higher School of Economics with state support for leading universities of the Russian Federation, project no. 5-100. Pushkar’s research was supported by the Russian Science Foundation (project no. 18-01-00461) and by the Simons Foundation.

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Authors and Affiliations

  1. National Research University Higher School of Economics, Moscow, Russia

    P. E. Pushkar & M. S. Temkin

  2. Independent University of Moscow, Moscow, Russia

    P. E. Pushkar

  3. Dartmouth College, Hanover, US

    M. S. Temkin

Authors
  1. P. E. Pushkar
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  2. M. S. Temkin
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Correspondence to P. E. Pushkar or M. S. Temkin.

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The authors declare that they have no conflicts of interest.

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Translated by I. Ruzanova

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Pushkar, P.E., Temkin, M.S. Bruhat Numbers of a Strong Morse Function. Dokl. Math. 106, 454–457 (2022). https://doi.org/10.1134/S1064562422700120

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  • DOI: https://doi.org/10.1134/S1064562422700120

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