# 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.

## References

1. [1]
A. Borel and J.C. Moore: “Homology theory for locally compact spaces”. Michigan Math. J., 7 (1960), 137–159.
2. [2]
J. Dixmier: “Homologie et cohomologie singulières”. Séminaire Henri Cartan, le année, 1948/1949, “Topologie algébrique”, expose No 5, 2e éd., Paris, Ecole normale supérieure, 1955.Google Scholar
3. [3]
M. Goresky and R. MacPherson: “Intersection homology theory”. Topology, 19 (1980), 135–162.
4. [4]
M. Goresky and R. MacPherson: “Intersection homology II”. Invent. Math., 72 (1983), 77–129.
5. [5]
P. Griffiths and J. Harris; “Principles of algebraic geometry”, New York, J.Wiley, 1978.
6. [6]
J.F.P. Hudson: “Piecewise linear topology”, New York, W.A. Benjamin, 1969, Mathematics lecture note series.
7. [7]
S. Lojasiewicz: “Triangulation of semi-analytic sets”. Ann. Scuola Norm. Sup. Pisa (3), 18 (1964), 449–474.
8. [8]
C. MacCrory: “Stratified general position”, in “Algebraic and geometric topology”, (Proc., Symp., Santa Barbara, in honor of Raymond L. Wilder, 1977), ed. by K.C. Millett, Berlin, Springer, 1978, Lecture notes in mathematics, No 664, 142–146.
9. [9]
J. Mather: “Stratifications and mappings”, in “Dynamical systems”, (Proc.,Symp., Univ., Bahia, Salvador, 1971), ed. by M.M. Pcixoto, New York, Academic press, 1973, 195–232.Google Scholar
10. [l0]
R. Thorn: “Espaces fibrés en spheres et carrés de Steenrod”. Ann. Sci. Ecole Norm. Sup., Paris, (3), 69 (1952), 109–182.