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Introduction to Piecewise Linear Intersection Homology

  • A. Haefliger
Part of the Modern Birkhäuser Classics book series (MBC, volume 50)

Abstract

Topological stratified pseudomanifolds. The definition is by induction on the dimension n. A stratified pseudomanifold X of dimension n is a topological space X with a filtration
$$ X = X_n \supset X_{n - 2} \supset X_{n - 3} \supset .... \supset X_1 \supset X_0 \supset X_{ - 1} = \varphi $$
by closed subspaces such that
  1. (i)

    Sn-k = Xn-k−Xn-k-1 is a topological manifold of dimension n-k (if Sn-k is non empty).

     
  2. (ii)

    X−Xn-2 is dense in X.

     
  3. (iii)
    local normal triviality : for each point xε Sn-k′, there is a compact stratified pseudomanifold L of dimension k-1
    $$ L = L_{k - 1} \supset L_{k - 3} \supset ... \supset L_0 \supset L_{ - 1} = \varphi $$
    and a homeomorphism h of an open nbhd U of x (called a distinguished nbhd of x) on the product \( Bx\mathop c\limits^ \circ L \) , where B is a ball nbhd of x in Sn-k, and \( \mathop c\limits^ \circ L \) is the open cone L×[0,∞[/(x,0)∼(x′,0) over L. Moreover h preserves the stratifications, namely h maps homeomorphically U⋂Xn-ℓ on \( Bx\mathop c\limits^ \circ \) Lk-ℓ-1 (by definition, the cone on the empty set is just a point).
     

Keywords

Singular Locus Hyperplane Section Euler Class Barycentric Subdivision Intersection Homology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston, Inc. 1984

Authors and Affiliations

  • A. Haefliger
    • 1
  1. 1.Faculté des Sciences Section de MathématiquesUniversité de GenèeveGenève 24Switzerland

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