Introduction to Piecewise Linear Intersection Homology

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


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).


Singular Locus Hyperplane Section Euler Class Barycentric Subdivision Intersection Homology 
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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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