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Morse Theory, Stratifications and Sheaves

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Handbook of Geometry and Topology of Singularities I

Abstract

After the local topological structure of stratified spaces was determined by R. Thom (Bull. Amer. Math. Soc., 75 (1969), 240–284) and J. Mather (Notes on topological stability, lecture notes, Harvard University, 1970) it became possible (see Kashiwara and Schapira, Sheaves on Manifolds, Grundlehren der math. Wiss. 292, Springer Verlag Berlin, Heidelberg, 1990; Goresky and MacPherson, Stratified Morse Theory, Ergebnisse Math. 14, Springer Verlag, Berlin, Heidelberg, 1988; Schürmann, Topology of Singular Spaces and Constructible Sheaves, Monografie Matematyczne 63, Birkhäuser Verlag, Basel, 2003) to analyze constructible sheaves on a stratified space using Morse theory. Although the detailed proofs are formidable, the statements and main ideas are simple and intuitive. This article is a survey of the constructions and results surrounding this circle of ideas.

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Notes

  1. 1.

    Characterized up to an additive constant by the condition that  = ι Y ω (interior product).

  2. 2.

    The imaginary part of the Fubini-Study metric.

  3. 3.

    Injectivity is an algebraic as well as a topological condition. The constant sheaf \({\mathbb Z}\) on a point is flabby and soft but not injective. It has an injective resolution \({\mathbb Z} \to \mathbb Q \to \mathbb Q/{\mathbb Z}\).

  4. 4.

    For more general rings it is necessary to replace R by an injective resolution, in which case the \( \operatorname {\mathrm {Hom}}\) above becomes a double complex and \(\omega ^{-r}_{BM}\) is defined to be the associated single complex.

  5. 5.

    If the coefficient ring is a field then a flabby or soft model of ω X suffices, see footnote 3.

  6. 6.

    By the universal coefficient theorem, the Euler characteristic may be computed with coefficients in any field. If X is a smooth n-dimensional manifold then \(H^i_c(X;{\mathbb Z}/(2)) \cong H_{n-i}(X;{\mathbb Z}/(2))\) by Poincaré duality hence χ c(X) = (−1)n χ(X).

  7. 7.

    In some situations, such as when a variety is stratified by the orbits of an algebraic group action, it is convenient to consider the category of perverse sheaves constructible with respect to a fixed stratification.

  8. 8.

    Similarly the Abelian category of sheaves is equivalent to the full subcategory of \(D^b_c(W)\) whose objects A satisfy: H r(A ) = 0 for r≠0.

  9. 9.

    Examples of non-hyperbolic fixed points include the point at infinity of the extension to \({\mathbb C}\mathbb P^1\) of the map zz + 1, for \(z\in {\mathbb C}\).

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Goresky, M. (2020). Morse Theory, Stratifications and Sheaves. In: Cisneros Molina, J.L., Lê, D.T., Seade, J. (eds) Handbook of Geometry and Topology of Singularities I. Springer, Cham. https://doi.org/10.1007/978-3-030-53061-7_5

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