CHARACTERISTIC CLASSES Jean-Paul BRASSELET CNRS Marseille

Preamble

These notes are prepared for a course given during the XVII Encontro Brasileira de Topologia, August 2nd to 6th, 2010 in PUC, Rio de Janeiro, Brasil. Notes of this course are partially taken from a book in preparation. Thanks to the reader for providing comments, critics etc... in order to improve the ﬁnal version.

Contents

1 Euler-Poincarécharacteristic 5 1.1 Combinatorial definition ...... 5 1.2 Pseudomanifolds ...... 6 1.3 Poincaréisomorphism ...... 8 1.4 The genus of surfaces ...... 9 1.5 Betti numbers ...... 9

2 Poincaré-HopfTheorem 12 2.1 The index of a vector field...... 12 2.2 The index - Definition by obstruction theory ...... 14 2.3 Relation with the Gauss map ...... 16 2.4 Poincaré-HopfTheorem ...... 17 2.4.1 The smooth case without boundary ...... 17 2.4.2 Consequences of Poincaré-HopfTheorem ...... 18 2.4.3 The smooth case with boundary ...... 19

3 Characteristic classes : the smooth case 21 3.1 Fibre bundles...... 21 3.1.1 Vector bundles ...... 22 3.1.2 Fibre bundles ...... 22 3.1.3 Examples of fibre bundles - real case ...... 23 3.1.4 Examples of fibre bundles - complex case ...... 24 3.2 General obstruction theory ...... 26 3.2.1 The difference cochain ...... 27 3.2.2 The obstruction class ...... 28

1 3.3 Case of the tangent bundle ...... 28 3.3.1 Index of a r-frame ...... 29 3.4 Applications: Stiefel-Whitney and Chern classes ...... 31 3.4.1 Stiefel-Whitney classes ...... 31 3.4.2 Chern classes ...... 32 3.5 Axiomatic deﬁnition ...... 33

4 Hirzebruch theory 34 4.1 The arithmetic genus ...... 34 4.2 The Todd genus ...... 34 4.3 The signature ...... 35 4.4 Hirzebruch Theory ...... 35 4.4.1 Hirzebruch Series ...... 35 4.5 Characteristic Classes of Manifolds ...... 35 4.6 The χy-characteristic ...... 36 4.7 Hirzebruch Riemann-Roch Theorem ...... 37

5 Singular varieties 38 5.1 Stratifications ...... 38 5.1.1 Whitney stratifications ...... 39 5.2 Poincaréhomomorphism ...... 39 5.2.1 Alexander isomorphism ...... 40 5.3 Poincaré-HopfTheorem: The singular case ...... 40 5.3.1 Radial extension process ...... 42 5.3.2 Poincaré-Hopf Theorem for singular varieties...... 44

6 Schwartz and MacPherson classes 47 6.1 Radial frames ...... 47 6.1.1 Global radial extension ...... 48 6.2 Schwartz classes ...... 50 6.3 Nash transformation ...... 50 6.4 Mather classes ...... 52 6.5 Euler local obstruction ...... 52 6.5.1 Properties of Euler local obstruction ...... 53 6.6 MacPherson classes ...... 53 6.7 Schwartz and MacPherson classes ...... 55 6.8 Schwartz-MacPherson classes for projective cones ...... 58 6.9 Schwartz-MacPherson classes of Thom spaces associated to embeddings . . 60

7 Other classes and comparisons 62 7.1 Fulton classes ...... 62 7.2 Milnor classes ...... 62 7.2.1 Description in terms of constructible functions ...... 63 7.3 Motivic Chern classes: Hirzebruch theory for singular varieties ...... 64 7.4 Verdier Riemann-Roch Formula ...... 66

2 Introduction

The Euler-Poincarécharacteristic is the first characteristic class that has been intro- duced. For a triangulated (possibly singular) compact variety X without boundary, it is defined, as X i χ(X) = (−1) ni , where ni is the number of i-dimensional simplices of the triangulation of X. It is also P i equal to (−1) bi where bi is the i-th Betti number of X, i.e. the rank of Hi(X). The Poincaré-Hopftheorem says that, if X is a compact manifold and v a continuous vector field with a finite number of isolated singularities ak with indices I(v, ak), then X χ(X) = I(v, ak) .

That means that the Euler-Poincarécharacteristic is a measure of the obstruction to the construction of a non-zero vector field tangent to X. Later on, Severi and Todd (1935) defined characteristic cycles of projective varieties using polar varieties. In his famous paper, Chern (1946) defined characteristic classes for hermitian manifolds in several ways, in particular as the measure of the obstruction to the construction of complex r-frames tangent to the manifold, generalising the Poincaré-Hopftheorem. The Chern classes are represented by algebraic cycles which coincide with Todd cycles. During several years, the attractiveness of the axiomatic properties of Chern classes caused the viewpoints of obstruction theory and polar varieties to be somewhat forgotten. It is interesting to see that these viewpoints came back on the scene with the question of defining characteristic classes for singular varieties. There are in fact various definitions of characteristic classes for singular varieties. In the real case, there is a combinatorial definition, which simplifies the problem. In the complex case, the situation is more complicated (and certainly more interesting !), due to the fact that there is no combinatorial definition of Chern classes. The obstruction theory point of view, in the smooth case, is based on the existence of the tangent bundle. If one wants to use the obstruction theory point of view in the singular case, one has to find a substitute to the tangent bundle. There are various candidates to substitute the tangent bundle and each of them leads to a different definition of Chern class for singular varieties. In particular, one has the following three substitutes. a) If X is a singular complex analytic variety, equipped with a Whitney stratification and embedded in a smooth complex analytic manifold M one can consider the union of tangent bundles to the strata, that is a subspace E of the tangent bundle to M. The space E is not a bundle but it generalises the notion of tangent bundle in the following sense: A section of E over X is a section v of TM|X such that in each point x ∈ X, then v(x) belongs to the tangent space of the stratum containing x. Such a section is called a stratified vector field over X. To consider E as a substitute to the tangent bundle of

3 X and to use obstruction theory is the M.H. Schwartz point of view (1965), for defining Chern classes of analytic complex varieties. b) A second possibility is to consider, for x singular point of X, the space of all possible limits of tangent vector spaces Txn (Xreg) where xn is a sequence of points in the regular part Xreg of X converging to x ∈ X. That point of view leads to the notion of Mather classes, which are an ingredient in the MacPherson definition for classes of algebraic complex varieties (1974). Another main ingredient for the definition of these classes is the notion of Euler local obstruction that we study in Section 6.5. c) A third possibility for a substitute to the tangent bundle is the following: Let us suppose that there exists a normal bundle N to X in M, that is the case of local complete intersections for example. Then, one can consider the virtual bundle TM|X \ N as a substitute to the tangent bundle of X. That point of view is the one of Fulton (1980). There are relations between the classes obtained by the previous constructions. First of all, the Schwartz and MacPherson classes coincide, via Alexander duality (1979, [B-S]). The relation between Mather classes on one side and Schwartz-MacPherson classes on the other side follows form the MacPherson’s definition itself: His construction uses Mather classes, taking into account the local complexity of the singular locus along Whit- ney strata. This is the role of the local Euler obstruction. A natural question arised to compare the Schwartz-MacPherson and the Fulton classes. A result of Suwa [Su1] shows that in the case of isolated singularities, the difference of these classes is given (up to sign) by the sum of the Milnor numbers in the singular points. It was natural to call Milnor classes the difference in the general case. This difference has been described by several authors using different methods (P. Aluffi, J.P. Brasselet-D. Lehmann-J. Seade-T. Suwa, A. Parusiński-P. Pragacz and S. Yokura). In the case of manifolds, the Todd genus and the L-genus are degree 0 elements of the Todd class and the Thom-Hirzebruch L-class. Both of them are related to the Chern class via the Chern roots and F. Hirzebruch gave a way to unify these three theories of characteristic classes by using the so-called multiplicative series. In the same way that the MacPherson construction generalises the Chern class, the Todd class and the Thom-Hirzebruch L-class have been generalised as natural transformations respectively by Baum-Fulton-MacPherson and by Cappell-Shaneson. The problem is that the three transformations are defined on different groups on the singular complex algebraic variety X: namely group of constructible functions, Grothendieck group of coherent sheaves, group of constructible self-dual sheaves. One way to unify the three theories, in the singular case, is to use the motivic theory and the Grothendieck relative group of algebraic varieties over X. That has been performed by J.-P. Brasselet, J. Schürmannand S. Yokura and is explained in the section 7.3.

The author thanks the Scientiﬁc and Organising Committees of the XVII Encontro Brasileira de Topologia para dar oportunidade de palestrar este curso. Aubagne, 26 de Junho 2010

4 1 Euler-Poincar´echaracteristic

In this section, the varieties we consider are possibly singular varieties.

1.1 Combinatorial definition History of characteristic classes begins with the discovery of the so-called Euler formula, by Leonhard Euler around 1750 : Let P be a 2-dimensional polyhedron in R3, homeomorphic to the sphere S2, one has k0 − k1 + k2 = 2 where k0 is the number of vertices in P , k1 is the number of segments and k2 the number of faces. That is the case for the tetraedron: 4 − 6 + 4, for the cube (with diagonals on the faces): 8 − 18 + 12. According to different authors, that formula was first proven by Euler himself, by Legendre in 1794 or by Cauchy. In fact, it seems that this formula was already known by R. Descartes (around 1620) and even by Archimedes. Simon Antoine-Jean Lhuilier, a Swiss mathematician, gave (in 1812) a slight generalization of Euler’s formula taking into account orientable 2-dimensional polyhedra with holes. The number g of holes is called genus. Lhuilier’s formula is

k0 − k1 + k2 = 2 − 2g, where g is the genus. Thus one obtains 0 for a torus-like polyhedron. For a general 2-dimensional polyhedron P in R3, the alternative sum

χ(P ) = k0 − k1 + k2 is called Euler characteristic of P . H. Poincaré[Po2] generalized the result in 1893 for finite polyhedra P of higher dimensions and proved the so-called Poincaré-HopfTheorem, which is the bridge to differential geometry. One defines

Definition 1.1 Let us denote by ki the number of i-dimensional simplices of a finite n-dimensional polyhedron P in Rm, the Euler-Poincarécharacteristic of the polyhedron P is defined by n X i χ(P ) = (−1) ki. i=0 Let us remind some elementary definitions:

5 Deﬁnition 1.2 A (ﬁnite) simplicial complex K is a collection of simplexes in some euclidean space such that

• if s ∈ K then every face of s belongs to K,

• if s, t ∈ K, then s ∩ t is either empty or is a common face of s and t.

Deﬁnition 1.3 Let us denote by K a (ﬁnite) simplicial complex in Rm. The union of simplexes in K is a compact subspace of Rm denoted by |K| = P and called geometric realisation of K, or polyhedron associated to K.

Deﬁnition 1.4 A topological space X is triangulable (or a polyhedron) if there exists a simplicial complex K and a homeomorphism h : |K| → X. Such a pair (K, h), or simply the simplicial complex K, is called a triangulation of X.

The Poincar´e’sresult is the following:

Theorem 1.5 (Poincar´e,[Po2]) Let us consider two triangulations (K1, h1) and (K2, h2) of a (compact) topological space X, then one has χ(K1) = χ(K2).

Theorem 1.5 implies that Euler-Poincar´echaracteristic is a topological invariant of the space X. The result makes sense for the following deﬁnition.

Definition 1.6 The Euler-Poincarécharacteristic of the triangulable space X, denoted by χ(X) is defined as χ(K) for a triangulation (K, h) of X.

Examples 1.7 The Euler-Poincar´echaracteristic of the sphere is χ(Sn) = 1 + (−1)n, of the 2-dimensional real torus T is χ(T ) = 0, of the pinched torus is χ(T ) = 1. The Euler-Poincar´echaracteristic of the complex projective space is χ(CPn) = n + 1.

1.2 Pseudomanifolds The spaces we will consider are pseudomanifolds. This corresponds to the spaces called n- circuit by Poincar´eand Lefschetz. In fact, the notion of pseudomanifold diﬀers according to the authors.

Definition 1.8 (Pseudomanifold - Combinatorial definition) One says that the polyhedron |K| is an n-pseudomanifold if the simplicial complex K satisfies the following properties:

(i) dim K = n, i.e. the maximal dimension of simplexes in K is n.

(ii) Each simplex is face of a n-simplex.

(iii) Each n − 1-simplex is face of exactly two n-simplexes.

The notion of “simplicial simple n-circuit” (Lefschetz [Le], Poincar´e) corresponds to the one of pseudomanifold with the following additional connexity property

6 (iv) The set of the n and n − 1-simplexes is connected.

The property means that |K| \ |K(n−2)| is connected. Equivalently, given two n simplexes σ and τ in K, there exists a sequence of n-simplexes σ = σ1, σ2, . . . , σr = τ such that σi ∩ σi+1 is an (n − 1)-simplex. If properties (i) to (iv) are veriﬁed, we will say that |K| is a simple n-pseudomanifold.

The topological deﬁnition of pseudomanifolds, which is equivalent to the combinatorial one in the case of triangulable topological space, goes as follows:

Definition 1.9 (Pseudomanifold - Topological definition) One says that the (paracompact, Hausdorff) topological space X is an n-pseudomanifold if there is a subset Σ ⊂ X such that:

(i0) dim X = n.

(ii0) X \ Σ is a n-topological manifold dense in X.

(iii0) dim Σ ≤ n − 2.

The property (iv) is equivalent to the following connexity property

(iv0) The set X \ Σ is connected.

If properties (i0) to (iv0) are veriﬁed, we will say that X is a simple n-pseudomanifold. In the triangulated case, one can take Σ = |K(n−2)|.

Example 1.10 The pinched torus, the suspension of the torus, a Thom space, a complex algebraic variety are examples of simple pseudomanifolds. Let K be a triangulation of a connected homology n-manifold (over Z), then |K| is an n-pseudomanifold.

Not all n-pseudomanifolds are homology n-manifolds. The pinched torus and the suspension of the torus are pseudomanifolds, they are not (homological) n-manifolds.

Proposition 1.11 An oriented simple n-pseudomanifold X admits a fundamental class [X] ∈ Hn(X).

Proof: Let us consider a triangulation (K) of X. It is easy to verify that the sum of oriented n-simplices is a cycle, called a fundamental cycle. Its homology class is the fundamental class of X.

7 1.3 Poincaréisomorphism Let us recall the definition: Definition 1.12 (Topological manifold) A Hausdorff space is called a (topological) m- m manifold if each point x in M admits a neighbourhood Ux homeomorphic to a ball B ⊂ m m R through a homeomorphism φ : Ux → B such that φ(x) = 0 and the boundary of Ux, called the link of x, is homeomorphic to the sphere Sm−1. Let us denote by M a m-manifold and by (K) a triangulation of M. A dual cell decomposition of M is obtained in the following way: Let us consider a barycentric subdivision (K0) of (K). The barycenter of a simplex σ ∈ K will be denoted byσ ˆ. Every simplex in K0 can be written as

(ˆσi1 , σî2 ,..., σîp ) 0 0 where σi1 < σi2 < ··· < σip . Here the symbol σ < σ means that σ is a face of σ . The dual cell of a simplex σ, denoted by d(σ), is the set of all (closed) simplexes τ in 0 (K ) such that τ ∩ σ = {σˆ}. That is the set of simplexes on the form (ˆσ, σî1 ,..., σîk ) with

σ < σi1 < ··· < σik . The dual cells satisfy the nice properties: Lemma 1.13 1. The dual cell is a cell, homeomorphic to a ball and its boundary is homeomorphic to the corresponding sphere. 2. If σ is a k-simplex, then d(σ) is a (m − k)-cell. 3. The set of dual cells provide a cell decomposition of M, called dual cell decomposition associated to the barycentric subdivision (K0) of (K). The unique intersection pointσ ˆ = d(σ) ∩ σ is the barycenter of σ that we will denote also sometimes by dˆ= dˆ(σ). Let us assume M = |K| oriented, that is all m-simplices are given a compatible orientation. One gives to every cell d(σ) the orientation such that orientation of σ followed by orientation of d(σ) is orientation of M. Let us ﬁx some notations: • We denote by d∗(σ) the elementary (D)-cochain whose value is 1 at the cell d(σ) and 0 at other cells of (D).

(K) • We denote by Ci the groups of simplicial K-chains with integer coefficients and i by C(D) the groups of simplicial D-cochains with integer coefficients. Let us consider a compact oriented m-dimensional manifold, then one has, for every k, a chain isomorphism: m−k (K) C(D) (M; Z) −→ Ck (M; Z), (1.14) that one defines on the elementary elements as d∗(σ) 7→ σ and extends linearly. The following Theorem is one of the possible forms of the Poincaréduality:

8 Theorem 1.15 (Poincar´eisomorphism) Let M be a compact oriented m-dimensional manifold, the morphism (1.14) induces, for every k, an isomorphism

m−k H (M; Z) −→ Hk(M; Z) , which is the cap-product with the fundamental class [M] ∈ Hm(M; Z).

1.4 The genus of surfaces Deﬁnition 1.16 The genus g of a connected surface is the integer representing the max- imum possible number of cuttings along closed simple curves without obtaining a discon- nected manifold.

Proposition 1.17 An orientable surface of genus g can be obtained from S2 by succes- sively attaching handles g times.

Among orientable surfaces, genus of the sphere is 0, genus of the torus is 1. We will see (Proposition 1.26) that one has χ(X) = 2 − 2g. Let us describe the “attaching process”: Let us consider a connected surface M and an embedding f : S0 × D2 → M \ ∂M. Image of f is a pair of disjoint disks in M. Cut 1 1 0 1 out interior of these disks and glue in the cylinder D × S by f|S ×S . One says that the resulting surface M 0 is obtained from M attaching an handle by f:

0 0 2 1 1 M = M \ Intf(S × D ) ∪f D × S .

Lemma 1.18 (see [Hirs]) Every nonorientable surface contains a M¨obiusstrip.

Theorem 1.19 (see [Hirs]) For a compact connected nonorientable surface without boundary the genus is the unique integer g such that M contains g but not g +1 disjoint M¨obius strips.

Proposition 1.20 A non-orientable surface of genus g ≤ 1 is diﬀeomorphic to the connected sum of g disjoint copies of RP2, real projective plane.

The real projective plane RP2 is a non-orientable surface of genus 1. One has χ(RP2) = 1. The Klein bottle B is a non-orientable surface of genus 2, one has χ(B) = 0. The sphere S2 with g real projective planes attached is a non-orientable surface of genus g. We will see (Proposition 1.26) that if X is a non-orientable surface of genus g, then one has χ(X) = 2 − g.

1.5 Betti numbers In 1871, Betti [Be] defined numbers relative to 3-dimensional compact manifolds without boundary and announced a duality property. A first statement of Poincaréduality was provided by Henri Poincaréin 1893 in terms of Betti numbers: The i-th and (n − i)-th Betti numbers of a closed (i.e. compact and

9 without boundary) orientable n-manifold are equal. In his 1895 paper “Analysis Situs” [Po3], Poincaréproved the theorem using a new tool: topological intersection theory. In his danish dissertation thesis, 1898, Poul Heegaard [He] (french translation in [HeF]) gives a counter-example to the version of Poincaréduality. The Heegard paper forced Poincaréto be more precise. In fact, Poincaréhad overlooked the possibility of the appearance of torsion in the homology groups of a space. According to Poincaré,the Heegard and Poincarédefinitions of Betti numbers (in fact definitions of homologies) are not the same (see NDLR, page 161 in [HeF]). The Betti definition does not take into account possibility to consider cycles with coefficients. In order to underline clearly this fact and to provide an indisputable proof, Poincaré wrote the first complement to Analysis situs [Po4]. In the first two complements to Analysis Situs [Po4, Po5], Poincarégave a new proof in terms of dual triangulations.

Let us denote by |K| a n -dimensional finite polyhedron and by Ci(K) the finitely generated free abelian group whose generators are (oriented) i-simplexes of the triangulation K. The boundary operator is classically defined as a complex map ∂i : Ci(K) → Ci−1(K). The subgroups of cycles and boundaries

Zi(K) = Ker[∂i : Ci(K) → Ci−1(K)] and Bi(K) = Im[∂i+1 : Ci+1(K) → Ci(K)] are finitely generated, as subgroups of a finitely generated group. The homology groups Hi(K, Z) = Zi(K)/Bi(K) are also finitely generated, as quotient group of a finitely generated one. One can write Hi(K, Z) = Fi(K) ⊕ Ti(K) where Fi(K) is the free subgroup and Ti(K) the torsion subgroup of Hi(K, Z).

Deﬁnition 1.21 The Betti numbers of |K| are deﬁned as

βi(K) = rk (Hi(K, Z)) = rk (Fi(K)).

Equivalently, one can deﬁne the Betti number βi(K) as the dimension of the vector space Hi(|K|; Q).

Noting that βi(K) = 0 if i > dim |K| = n, one has the Poincar´eTheorem:

Theorem 1.22 (Poincar´eTheorem) [Po3] Let |K| be a ﬁnite polyhedron in Rm, with Betti numbers βi(K), one has

n X i χ(K) = (−1) βi(K). i=0

Proof: From the long exact sequence

∂i+1 ∂i · · · −→ Ci+1(K) −→ Ci(K) −→ Ci−1(K) −→ · · ·

10 one deduces the following equalities (where ni = rk (Ci(K)) and n = dim(|K|))

n n n n X i X i X i X i (−1) ni = (−1) (dim Zi +dim Bi−1) = (−1) (dim Zi −dim Bi) = (−1) βi(K). i=0 i=0 i=0 i=0 The ﬁrst one comes from the short exact sequence

0 → Zi → Ci → Bi−1 → 0 , the second one because B−1 = ∅ and Bn = ∅ and the third one from the Deﬁnition 1.21. The Theorem follows. Alexander [Al] proved in 1915 that two triangulations (K, h) and (K0, h0) of the same 0 topological space X have same Betti numbers βi(K) = βi(K ) for every i. One can deﬁne βi(X) as being βi(K) for any triangulation (K, h) of X and one has

n X i χ(X) = (−1) βi(X). i=0 This result proves that each Betti number is a topological invariant. In that sense, it is more precise than the Poincar´eTheorem 1.5, which globally proves invariance of Euler- Poincar´echaracteristic only.

Theorem 1.23 The Betti numbers of a compact orientable n-manifold M satisfy

βi(M) = βn−i(M) for i = 0, 1, . . . , n.

Corollary 1.24 If M is a compact orientable n-manifold with odd n, then χ(M) = 0.

Examples 1.25 If n is odd, the Euler-Poincar´echaracteristic of the sphere Sn, the real projective space RPn, a compact hypersurface in Rn+1, are zero.

Proposition 1.26 If X is an orientable (connected) surface of genus g, then β0(X) = 1, β1(X) = 2g and β2(X) = 1. One has χ(X) = 2 − 2g. In X is a non-orientable surface of (non-orientable) genus g, then β0(X) = 1, β1(X) = g − 1 and β2(X) = 0. One has χ(X) = 2 − g.

11 2 Poincar´e-HopfTheorem

The Poincaré-HopfTheorem is important for many points of view: that is the first result that links two invariants from topology and differential geometry. The Poincaré-HopfTheorem has been proved by Poincaré[Po1] in 1885, in the 2- dimensional case, and by Hopf in 1927 [Ho] for higher dimensions. In between, partial results had been proved by Brouwer and Hadamard. This result is the first apparition of Euler-Poincarécharacteristic in differential topology, out of combinatorial topology. It seems strange that Poincaréextended the notion of Euler-Poincarécharacteristic from dimension 2 to the general case but proved the Poincaré-HopfTheorem in dimension 2 only without extending it in the general dimension. The meaning of Poincaré-HopfTheorem is that Euler-Poincarécharacteristic is a measure of the obstruction to constructing a continuous vector field tangent to the considered manifold, without singularity. One of the motivations of the Poincaré-HopfTheorem is the study of differential equations in terms of integral curves of an appropriate vector field. The singular points of the vector field are points of equilibrium in dynamical systems. That is the reason for which Poincaré-HopfTheorem has many applications: mathe- matical economics, optimisation of communication systems, electrical engineering, applied probability (cooperative dynamical systems), statistical complexity, particle physics (elec- tromagnetic fields), structure of materials: stability of molecular complexes in chemistry, crystallography, graphics applications, astrophysics: magnetic fields, etc... The interested reader should experience to search for “Poincaré-HopfTheorem” on his/her favorite web search engine. The first part of the chapter is devoted to various definitions of the index of a vector field in an isolated singularity.

2.1 The index of a vector field. In this section, one gives different ways to define the index of a vector field in an isolated singular point. The obstruction theory definition will be useful for the following and provides a geometrical meaning to characteristic classes. One can find generalisation of the theory to non-isolated singularities in [BLSS2] (see [BSS] for a systematic study). The index of a vector field at an isolated singularity can be defined in various ways. We limit ourself to the “classical” ones (see [B3, BSS]). In a first step, we consider vector fields in Euclidean space, then we will define the index for vector fields tangent to a manifold. n Let Ω be an open subset in R with coordinates (x1, . . . , xn). Let

12 n X v = fi ∂/∂xi i=1 be a vector field on Ω. The vector field is said to be continuous, smooth, analytic, according as its components {f1, . . . , fn} are continuous, smooth, analytic, respectively. A singularity a of v in Ω is a point where all of its components vanish, i.e., fi(a) = 0 for all i = 1, . . . , n. The singularity is isolated if at every point x near a there is at least one component of v which is not zero. Let v be a continuous vector field on Ω with an isolated singularity at a, and let B(a) be a small ball in Ω around a so that there is no other singularity of v within B(a). Let us define the Gauss map

∼ n−1 n−1 γ : ∂B(a) = S(a) = S −→ S by γ(x) = v(x)/kv(x)k.

Deﬁnition 2.1 The (local) index of v at a, denoted by I(v, a), is the degree of the Gauss map γ : Sn−1 → Sn−1. The local index does not depend on the choice of the small ball B(a), on the choice of coordinates nor on the choice of orientations.

Remark 2.2 In the following, we will use also a different kind of singularities for a vector field, that M.-H. Schwartz called second type singularities. Let us introduce these singularities. Given a vector field v defined on the boundary S(a) of the ball B(a) of radius 1, centred in a, there are many ways to extend the vector field inside B(a). Two are the most natural. Let us denote by Sε(a) the sphere of radius ε, 0 < ε ≤ 1. If x ∈ S(a) the vector v(εx) at the point εx ∈ Sε(a) is defined either as v(εx) = εv(x) or as v(εx) = v(x). In the first case, the vector field v will be 0 in a, that is the already defined singularity type. We will call it, according to M.-H. Schwartz, singularity of first type. In the second case the extension is not defined at a (see [Sc4]), but it defines a cycle n n κ(v) in the fibre TaR of the tangent bundle to R . We will call it, again according to M.-H. Schwartz, singularity of second type. Whatever the type of singularity, the index I(v, a) of v at the isolated singularity a is well defined by the Definition 2.1.

Proposition 2.3 Let us consider a second type singularity, then the index of the cycle n κ(v) in the punctured ﬁbre TaR \{0} is equal to I(v, a).

n Proof: Let us denote by s0 the zero section of the tangent vector bundle T R . The n × n n tangent bundle T R is trivial over B(a), as well as the bundle T R = T R \ Ims0 (not × n n ∼ n anymore a vector bundle). The fibre of T R at a is TaR \{0} = R \{0} and, restricted to B(a), the bundle is homeomorphic to B(a)×(Rn \{0}). The vector field v defines a section of T ×Rn over S(a) whose image by the second projection B(a) × (Rn \{0}) → Rn \{0} is equal to κ(v), by definition. One concludes by the Definition 2.1.

13 Let us consider now a n-dimensional smooth manifold M. A continuous vector field on M is a section of its tangent bundle TM (see 3.1). Giving a local chart (Ua, φ) on M, n where φ : Ua → B , a vector field on M is locally expressed as above: Let us denote by xi = xi ◦ φ the coordinate functions of φ, i.e. the local coordinates in Ua. We denote by ∂/∂xi the tangent vector at x defined by

∂ ∂ −1 (h) = (h ◦ φ )|φ(x) ∂xi ∂xi for a C∞ function h : M → R.

Deﬁnition 2.4 Let us denote by x = (x1, . . . , xn) the local coordinates of the manifold M in the open neighborood Ua, a vector ﬁeld v can be written in terms of the basis ∂/∂xi of the tangent vector space TxM n X ∂ v = f . (2.5) i ∂x i=1 i

The functions (f1, . . . , fn) are called coordinates of the vector v in Ua. The vector ﬁeld is said to be continuous, smooth, analytic, according as its components {f1, . . . , fn} are continuous, smooth, analytic, respectively.

n For simplicity, in the following, we will identify coordinates in Ua and B , omitting φ and we will denote xi for xi. A singularity (“first type singularity”) a of the vector field v is a point in which all coordinate fi vanish. One can also define “second type singularities” of a vector field v in the same way than in the Euclidean situation. The index of the vector field v at a is well defined in both cases as the degree of the Gauss map −1 n ∼ n−1 n−1 γ : ∂φ (B ) = S −→ S

2.2 The index - Definition by obstruction theory The index can be also defined in the following way: Let M be a differentiable manifold of dimension n. The tangent bundle to M, denoted by TM, is a real vector bundle (see section 3.1) of rank n, whose fibre in a point x of M is the tangent space to M at x, n denoted by Tx(M) and is isomorphic to R . The vector bundle TM is locally trivial, i.e. there is a covering of M by open subsets {Ua} such that the restriction of TM to each n Ua is homeomorphic to Ua × R . Let us denote by s0 the zero section of TM, we will consider the bundle (not any more × × ∼ n a vector bundle) T M = TM \ s0(M). Its fibre in a point x ∈ M is Tx M = R \{0}. Let us consider a ball B(a) centred in a, contained in an open chart Ua over which TM is trivial and sufficiently small so that a is the only singular point of v in B(a). One can think of B(a) as an n-cell, in view of the generalisation we will perform later (3.15). The vector field v defines a section of TM without zero over S(a) = ∂B(a), hence a map

∼ n−1 v × ∼ n pr2 n S(a) = S −→ T M|Ua = Ua × (R \{0}) −→ R \{0} (2.6)

14 where pr2 is the second projection. One obtains a map n−1 ∼ pr2◦v n S = ∂B(a) −→ R \{0} n hence an element λ(v, a) in πn−1(R \{0}). One knows that this homotopy group is Z. The generator +1 can be interpreted in the following way:

Let us consider the radial vector field vrad, that is the vector field pointing outwards Pn n the ball B(a) along S(a), image of the vector field i=1 ∂/∂xi in R . Image of vrad in n n−1 n−1 R \{0} is S and the map pr2 ◦ vrad is the identity of S . The previous construction shows that λ(v, a) corresponds to the index I(v, a) by the n ∼ isomorphism πn−1(R \{0}) = Z. By classical homotopy theory, the map

n−1 ∼ pr2◦v n S = ∂B(a) −→ R \{0} extends to a map n B(a) −→ R \{0} n if and only if the element λ(v, a) is zero in πn−1(R \{0}), i.e. the vector ﬁeld v extends within the ball B(a) if and only if the index I(v, a) is zero.

∼ n−1 n ∂B(a) = S −−→ R \{0}    ? % (2.7) y ∼ n B(a) = B

That construction is the basis of obstruction theory, it will be generalised in chapter 3.

Lemma 2.8 There is a homotopy

× × ψ : S(a) × [0, 1] → T (B(a)|S(a)) ⊂ T (M|S(a)) such that

∂Imψ = v(S(a)) − I(v, a) · pr2 ◦ vrad(S(a)) (2.9)

Proof: Let us suppose without loss of generality that v is an unitary vector field on × S(a). For t 6= 0, let us define in the fibre Ttx(M)

ψt(x) = ψ(t, x) = the unitary vector parallel to v(x) at the point tx.

× If t goes to 0, then ψ0(x) is the unit vector in T0 (M) parallel to v(x) and with origin 0. × Therefore ψ0(S(a)) is a cycle in the fiber T0 (M) whose index is I(v, 0). In the case of the radial vector field vrad on S(a), the cycle is the projection pr2 ◦ vrad(S(a)) over the fibre at 0 (by local triviality of the bundle) and it has index I(vrad, 0) = 1. One concludes the Lemma.

15 2.3 Relation with the Gauss map

Let N a compact k-manifold with boundary in Rk. The Gauss map

k−1 g : ∂N → S assigns to each x ∈ ∂N the outward unit normal vector at x.

Lemma 2.10 (Hopf) ([Mi1], §6, Lemma 3) If v is a smooth vector ﬁeld on N with isolated singularities ai and v points outward of N along the boundary, then the sum of P k−1 indices I(v, ai) equals the degree of the Gauss mapping from ∂N to S .

Proof: For each singular point ai one considers a small (closed) ball B(ai) with center ai and which do not intersect each other. The vector ﬁeld v has no singularity in S W = N \ i B(ai). k Let us consider a compact n-manifold without boundary M ⊂ R . Let Nε denote the closed ε-neighbourhood of M (i.e. the set of all x ∈ Rk with kx−yk < ε for some y ∈ M). For ε suﬃciently small, Nε is a smooth manifold with boundary.

Theorem 2.11 Let v be a vector ﬁeld with isolated singularities ai, on a compact manifold k P without boundary M ⊂ R , the index sum I(v, ai) is equal to the degree of the Gauss mapping k−1 g : ∂Nε → S , k where Nε denotes the closed ε-neighbourhood of M in R .

We reproduce the proof due to Milnor [Mi1], §6, Theorem 1. That proof has been delivered in December 1963 in lectures in University of Virginia. The procedure used by Milnor is the same as the one developed independently and at the same time by M H Schwartz [Sc1], in her definition of radial extension in the framework of stratified singular varieties. More precisely, the idea is to extend a vector field v defined on the manifold M with index I(v, a; M) at the isolated singularity a, as a vector field w in the ambient space Rk that has also an isolated singularity at a with the same index I(w, a; Rk) = I(v, a; M). The principle is to sum the parallel extension of v in a neighbourhood of a with a transversal vector field.

Proof: For x ∈ Nε, let r(x) be the closest point of M. The vector x − r(x) is perpendicular to the tangent space of M at r(x), for otherwise, r(x) would not be the closest point of M. If ε is suﬃciently small, then the restriction r(x) is smooth and well deﬁned. We consider the squared distance function (for the Euclidean metric in Rk):

φ(x) = kx − r(x)k2 whose gradient vector ﬁeld is

gradφ(x) = 2(x − r(x)).

16 On one hand, the gradient vector field is a vector field defined in Nε that is zero along M, that is transverse to ∂Nε going outward and that increases with the distance to M. For −1 2 each point x at the level surface ∂Nε = φ (ε ), the outward unit normal vector, called transversal vector, is given by

g(x) = gradφ(x)/kgradφ(x)k = (x − r(x))/ε.

On the other hand, in each point x ∈ Nε, the vector v1(x) = v(r(x)) is a parallel extension of v. Extend v to a vector ﬁeld w on the neighbourhood Nε by setting

w(x) = (x − r(x)) + v1(x).

The vector ﬁeld w points outward along the boundary ∂Nε, since the inner product w(x) · g(x) is equal to ε > 0. In fact w vanish only at the zeros of v in M. That is clear because the two summands (x − r(x)) and v1(x) are orthogonal. Now, the index of w at the zero a, computed in Rk is equal to the index of v at a, computed in M and, according to the Lemma 2.10, the index sum P I(v, a) is equal to the degree of g which proves the theorem. P The Theorem is another way to see that if M is compact, the sum I(v, ai) for all singularities of v does not depend on v. We will see (Theorem 5.12) that, with suitable vector ﬁelds, the result extends to the case of singular variety M.

2.4 Poincaré-HopfTheorem There are many ways to prove Poincaré-HopfTheorem. They correspond to the different viewpoints and definitions of the index. The interested reader can consult [Li], [Mi1] (Hopf and Gauss map), [GP] (Lefschetz fix points theory), [Hirs] (Intersection numbers).

2.4.1 The smooth case without boundary Theorem 2.12 [Poincaré-HopfTheorem] Let M be a compact differentiable manifold, and let v be a continuous vector field on M with finitely isolated singularities ai. One has X χ(M) = I(v, ai) i

Proof: Firstly we prove the Theorem in the orientable case, then in the non-orientable case. We will follow the Milnor proof which is close to the generalisation to singular varieties that we will provide in the next chapters. 1) Orientable case. The idea of the proof is the following: In a first step, one shows that the sum of indices of a continuous tangent vector field with isolated singularities does not depend of the choice of the vector field. The second step of the proof consists in describing a particular vector field for which the sum of indices is equal to χ(M). For the first step, Theorem 2.11 provides directly the result.

17 For the second step, such a vector field is given for instance by the gradient field associated to a Morse function. Another possibility is to consider the Hopf vector field H of which we recall the construction (see [Ste], p. 202). Let us consider a triangulation K of M and a barycentric subdivision K0 of K. The Hopf vector field will be tangent to simplexes of K0, with a singularity in every vertex of K0, i.e. in every barycenter of K. On every 1-simplex [ˆσ, τˆ] of K0, whereσ ˆ andτ ˆ are barycenters of σ and τ, and σ < τ, the vector field H is going in the direction fromσ ˆ toτ ˆ. For example it is going outward all vertices of K. The Hopf vector field H has a singularity of index (−1)i in the barycenter of every Pn i i-simplex of K. The sum of indices of H in all singularities is i=0(−1) ki where ki is the number of i-dimensional simplexes of K, so it is equal to χ(M). 2) The non-orientable case: Let us consider the oriented double covering π : Mf → M. On one hand, if v is a continuous vector field on M with isolated singular points ai of index I(v; ai), then on can define a liftingv ˜ of v which is a continuous vector field on Mf with isolated singular j j points ai , j = 1, 2 such that π(ai ) = ai. As π is a local homeomorphism, one has j P j P I(v; ai ) = I(v; ai) for j = 1, 2. One obtains i,j I(v; ai ) = 2 i I(v; ai). On the other hand, one has χ(Mf) = 2χ(M) (use suitable triangulations). One conclude the Poincaré- Hopf Theorem :

X j X χ(M) = 1/2 · χ(Mf) = 1/2 I(v; ai ) = I(v; ai). i,j i

2.4.2 Consequences of Poincar´e-HopfTheorem As an important consequence of the Poincar´e-HopfTheorem, one has the following

Corollary 2.13 Let M be a compact smooth manifold, if χ(M) 6= 0, then any continuous vector ﬁeld tangent to the manifold M admits at least a singular point. Reciprocally, every compact manifold such that χ(M) = 0 admits a continuous tangent vector ﬁeld without singularities.

The unitary sphere Sn with odd n satisfies χ(Sn) = 0 and admits continuous tangent vector fields without singularities. If n is even, χ(Sn) = 2 and in that case every continuous vector field tangent to Sn admits at least one singularity.

Corollary 2.14 Every compact odd dimensional manifold admits a continuous tangent vector ﬁeld without singularity.

The torus and the Klein bottle are the only one compact 2-dimensional surface ad- mitting a non-zero continuous tangent vector ﬁeld.

Lemma 2.15 For even-dimensional hypersurfaces, the Euler-Poincar´echaracteristic χ(M) equals twice the degree of the Gauss map γ.

18 Proof: Take the projection π : Sn → RPn and a regular value p ∈ RPn of the composed map π ◦ γ : M → RPn. Take a differentiable vector field w on Sn with isolated singularities in {a, b} = π−1(p) of indices +1. The vector field v on M such that −1 v(x) = w(γ(x)) has a finite number of isolated singularities {a1, . . . , ar} = γ (a) and −1 Pr Ps {b1, . . . , bs} = γ (b). One one hand, one has deg(γ) = i=1 I(v; ai) = j=1 I(v; bj), on Pr Ps the other hand χ(M) = i=1 I(v; ai) + j=1 I(v; bj). That gives the Lemma.

2.4.3 The smooth case with boundary Let M be an oriented manifold with boundary, one has a similar theorem:

Theorem 2.16 [Poincaré-HopfTheorem with boundary] Let M be a compact manifold with boundary ∂M embedded in an oriented differentiable manifold N. Let v be a non- singular continuous vector field tangent to N, strictly pointing outwards (resp. inwards) M along the boundary ∂M. Then:

1. v can be extended to the interior of M as a vector ﬁeld tangent to M with ﬁnitely many isolated singularities ai. 2. The total index of v in M is independent of the way we extend it to the interior of M. In other words, the total index of v is fully determined by its behaviour near the boundary.

3. If v is everywhere transverse to the boundary and pointing outwards from M, then one has X χ(M) = I(v, ai). (2.17) i If v is everywhere transverse to ∂M and pointing inwards then X χ(intM) = χ(M) − χ(∂M) = I(v, ai). (2.18) i

Proof: The first statement is proved by obstruction theory (section 2.2). The vector field can be extended without singularities to the (n − 1)-skeleton of M. Then we extend it to the n-cells introducing (if necessary) a singular point for each n-cell. The second statement is also a general result in obstruction theory, that can be obtained (for instance) as a consequence of statement 3 (see also 3.3 and [Ste]). A proof of the third statement goes in the following way: Like in Theorem 2.11, on consider the closed ε-neighbourhood of M, denoted by Nε. If the vector field is pointing outward along ∂M, then it can be extended over the neighbourhood Nε so that the extended one points outward along ∂Nε. The extension w is defined as before by w(x) = (x − r(x)) + v(r(x)) and is a continuous vector field near ∂M. In this case, Nε is not necessarily of class C∞, but only a C1-manifold. Nevertheless, the same argument as in the case “without boundary” can be carried out (see [Mi1] §6), that gives (2.17). If the vector field is pointing inward along ∂M, one can extend v inside M with finitely many isolated singularities ai of index I(v, ai).

19 One proceeds to the following construction: the boundary ∂M admits a neighbourhood ∂M × [0, 1] in M and one can extend this neighbourhood as ∂M × [0, 2]. Let us call M 0 the new manifold M ∪ (∂M × [0, 2]). One has χ(M 0) = χ(M) and ∂M 0 ∼= ∂M. Let us call C the collar ∂M × [1, 2]. One has χ(C) = χ(∂M). At the level C1 = ∂M ×{1}, one has the vector field v pointing inward M and outward 0 0 C. At the level C2 = ∂M × {2}, one considers any vector field v pointing outward M along ∂M 0. Let us call w the vector field defined on ∂C which is equal to v and v0 on C1 and C2 respectively. The vector field w is defined on the boundary of C and pointing outward C along the boundary. By (2.17) on C, one can extend w inside C with finitely many isolated singularities bj and one has X χ(C) = χ(∂M) = I(w, bj). j

On M 0 one consider the vector ﬁeld v0, which is v on M and w on C. It has isolated 0 0 singularities ai and bj and it is pointing outward M . Again one can apply (2.17) (on M ) and one has

0 X X X χ(M) = χ(M ) = I(v, ai) + I(w, bj) = I(v, ai) + χ(∂M) i i and the result.

Corollary 2.19 Let us suppose M is odd-dimensional, then

χ(∂M) = 2 · χ(M)

Proof: Let us denote by v a vector ﬁeld pointing outwards D along the boundary, like in 2.17 and let us consider the vector ﬁeld w = −v. Then, w has same singularities than v and, as M is odd-dimensional, in each singularity ai, one has I(w, ai) = −I(v, ai). Equations 2.17 and 2.18 provide the result.

Corollary 2.20 Let us denote by M ⊂ Rk a compact manifold without boundary in an k odd-dimensional Euclidean space and by Nε the closed ε-neighbourhood of M in R . Then one has χ(∂Nε) = 2 · χ(M)

20 3 Characteristic classes : the smooth case

In 1935 and independently, Stiefel, who was student of Hopf, and Whitney defined characteristic classes in cohomology for real manifolds. Stiefel considers the obstruction point of view (for the construction of r-frames tangent to the manifold), computing homotopy groups of so-called Stiefel manifolds. Whitney considers sphere bundles on a manifold M and defines cohomology classes with coefficients in Z/2 = Z/2Z. The Stiefel and Whitney methods are similar and represent the basis of obstruction theory. We call Stiefel-Whitney classes of a vector bundle or of the associated sphere bundle, the classes obtained in that way. In 1942 Pontrjagyn defined classes for Grassmannian manifolds, using a decomposition of these manifolds in terms of Schubert varieties, due to Ehresmann. In his fundamental 1946 paper [Ch], Chern gave several constructions of characteristic classes for Hermitian Manifolds. The paper provides basement for the relationship between obstruction theory, Schubert varieties, differential forms, transgression, etc... We will briefly recall some of these definitions, either because they extend to the case of singular varieties, or because they will be useful for the following. Contribution of Wu Wen Tsünin the history of characteristic classes is important. Among results, he proved the product formula for Stiefel-Whitney and Chern classes, he gave a simple formulation of the decomposition of the Grassmann manifold of oriented vector subspaces and he extended the definition of Chern classes for any complex vector space on any finite simplicial complex. As it happens often in Mathematics, one object, here the Chern classes, has (at least) two definitions: the geometric definition allows to understand the signification of classes, but it is difficult to proceed to effective computations in this context. The axiomatic definition provides easy ways to compute effectively the classes but is less suitable to understanding the origin and the meaning of the classes (see [MS, Hu]). A trivial bundle is induced from a map to a point, so all its characteristic classes (except the zero dimensional one) should be zero. More generally, equality of all characteristic classes of two bundles is a necessary (and in some circumstances sufficient) test for their equivalence. That is one of the important uses of characteristic classes. The interested reader will find all wished references in the Dieudonnébook [Di], 3,IV.

3.1 Fibre bundles.

In this section, we will denote by K either the real field R or the complex field C. We provide elementary definitions and properties of vector and fibre bundles, as well as a

21 series of examples in the real and complex situations. The reader will ﬁnd in the literature suitable references for more deﬁnitions and properties (see for instance [Hirz] and [Hu]).

3.1.1 Vector bundles

Definition 3.1 A vector bundle E, over the field K, with base X and rank n is a topological space E, with a continuous map π : E → X, the projection, such that for every −1 point x ∈ X, the fibre Ex = π (x) is a vector space of rank n over K. A vector bundle satisfies the local triviality condition: for every point x ∈ X, there is −1 n an open neighbourhood Ux in X and a homeomorphism φ : π (Ux) → Ux × K which −1 n induces for every y ∈ Ux an isomorphism π (y) → K . A trivial bundle is a bundle for which one has “global” triviality, i.e. one can take Ux = X in the previous condition. Given a vector bundle E over X, one define in a natural way the dual vector bundle E∗ and the bundle of k-vectors ΛkE whose fibres are respectively (Kn)∗ and ΛkKn. Vector bundles are special cases of fibre bundles that we recall now.

3.1.2 Fibre bundles Let F a topological space and G a topological group which acts effectively and continuously on F . That means there is a continuous map G × F → F such that one has g1 · (g2 · a) = (g1g2) · a for g1, g2 ∈ G and a ∈ F , and e · a = a if e is the identity element in G. The action is effective means that if g · a = a for some a ∈ F then g = e. Definition 3.2 A topological space E with a continuous projection π : E → X, is called a fibre bundle with fibre F and structure group G if G acts effectively and continuously on F and there are a system of coordinates (Ui, φi) on X and continuous functions gij : Ui ∩ Uj → G such that: −1 •{Ui} is an open covering of X and φi : π (Ui) → Ui × F is a homeomorphism identifying π−1(x) with the fibre {x} × F ,

−1 • (φi ◦ φj )(x, a) = (x, gij(x) · a) for all x ∈ Ui ∩ Uj and a ∈ F .

The fonctions gij are called transition functions. They satisfy

gij ◦ gjk ◦ gki = id for all i, j, k, hence, they define a cocycle in Z1(X,G), then an element in H1(X,G). It is well known that isomorphism classes of fibre bundles over X with fibre F and structural group G are in a one-one correspondence with the elements of H1(X,G). The trivial bundle corresponds to the element 1 ∈ H1(X,G). Fibre bundles in the same isomorphism class ξ ∈ H1(X,G) are said associated bundles. The fibre bundle is said differentiable if X is a differentiable manifold and G a real Lie group, the gij being differentiable functions. The fibre bundle is said complex analytic if X is a complex manifold and G a complex Lie group, the gij being holomorphic functions. A section of the fibre bundle E is a continuous application s : X → E such that, for −1 every point x ∈ X, one has s(x) ∈ Ex = π (x).

22 3.1.3 Examples of fibre bundles - real case In order to provide examples of real vector bundles and fibre bundles, we will use the following spaces: The real projective space RPn is the space of lines through the origin of Rn+1. The n n Grassman manifold Gr(R ) is the space of all vector subspaces of dimension r of R . ∞ Let R be the vector space of all infinite sequences (x1, x2,...) whose elements xi are real numbers, a finite number of them being nonzero. The infinite Grassmannian manifold ∞ ∞ Gr(R ) is the set of all r-dimensional subspaces in R , i.e. the direct limit of the natural sequence of inclusions

r r+1 r+2 Gr(R ) ⊂ Gr(R ) ⊂ Gr(R ) ⊂ · · ·

∞ We consider on Gr(R ) the topology for which closed subsets are those whose intersections r+k with all Gr(R ) are closed. Examples of real vector bundles are given by:

1. the tangent bundle TM to a diﬀerentiable manifold M. That is the set of all pairs (x, v) such that x ∈ M and v is a vector tangent to M at the point x, i.e. an element of TxM. If M is an n-manifold, then TM is a real vector bundle with rank n over M, the ﬁbre is Rn. In particular, one has the bundle T Sn tangent to the sphere Sn, that is a trivial bundle if n = 1, a non trivial bundle if n = 2. One has also the bundle T RPn tangent to RPn.

2. the normal bundle to a diﬀerentiable n-manifold M embedded in Rn+k. That is the set of all pairs (x, v) ∈ M × Rn+k such that v is orthogonal to the tangent space ∼ n n+k ∼ n+k TxM = R in Tx(R ) = R .

n n 3. the canonical bundle over RP also called tautological bundle and denoted by γ1 :

n n γ1 → RP (3.3)

n This line bundle is the set of all pairs {(λ, v)} where λ is an element of RP , i.e. a line passing through the origin of Rn+1 and v a vector of λ. The canonical bundle is not trivial, and this fact is the basis for the axiomatic deﬁnition of characteristic classes.

n n 4. the canonical bundle γr over the Grassman manifold Gr(R ). That is the set of all n pairs {(P, v)} where P is an element of Gr(R ) and v a vector in P . One has the bundle projection n n γr → Gr(R ) n and γr is a vector bundle with rank r. The bundle is also called universal bundle for vector bundles of rank r. That means that every bundle ξ with rank r over a (paracompact) topological space X is iso- ∗ n n morphic to f (γr ) for some f : X → Gr(R ) with suﬃciently large n.

23 5. the universal bundle ∞ γr → Gr(R ) ∞ set of all pairs {(P, v)} where P is an element of Gr(R ) and v a vector of P . It is universal for all rank r-vector bundles. ∞ In the case r = 1, that is the bundle γ1 → RP .

n n 6. the Stiefel manifold, denoted by Vr(R ) is the set of r-frames in R , that is the set n of ordered r-uples (v1, . . . , vr) of r linearly independent vectors in R . (see Steenrod 0 [Ste] where this manifold is denoted by Vr,n). n ∼ One has a homotopy Vr(R ) = Vr,n = O(n)/O(n − r). n The natural map Vr,n → Gr(R ) is a principal fibre bundle, i.e. the fibre O(r) coincides with the structural group. That is an universal bundle for fibre bundles whose basis has dimension ≤ n − r − 1. n n n The vector bundle γr → Gr(R ) is a bundle associated to Vr,n → Gr(R ) with fibre Rr.

7. the bundle Vr(TM) of r-frames tangent to a n-differentiable manifold M, i.e. the set of all pairs (x, (v1, . . . , vr)) where x is a point of M and (v1, . . . , vr) is a r-frame in the fibre TxM over x. That is the fibre bundle over M whose fibre at x is the n manifold Vr(TxM) of all r-frames in TxM. The fibre is the Stiefel manifold Vr(R ). Note that a section of this bundle in Stiefel manifolds is a r-uple of linearly independent sections of the vector bundle TM.

3.1.4 Examples of ﬁbre bundles - complex case

n One considers the complex projective space CP whose homogeneous coordinates will be denoted by (x0 : x1 : ... : xn). The projective space is covered by open subsets {Ui}i=0,...n n homeomorphic to C and whose coordinates are (x0, x1, . . . , xi−1, 1, xi+1, . . . , xn). n ∞ We will consider the complex Grassmannian manifolds Gr(C ) and Gr(C ) in a similar way than in the real case.

1. the complex tangent bundle TM to a complex n-dimensional manifold M is a ﬁbre n bundle over M. Each ﬁbre TxM has a complex structure and is isomorphic to C . In particular, one has the tangent bundle T CPn to CPn.

n n 2. the canonical bundle γ1 over CP . also called tautological or universal bundle and denoted by O(−1) in algebraic geometry:

n n γ1 → CP (3.4)

This line bundle is the set of all pairs {(λ, v)} where λ is an element of CPn, i.e. a complex line passing through the origin of Cn+1 and v a vector in λ. That is the ﬁbre over λ is the line λ.

n n n+1 γ1 = {(λ, v) ∈ CP × C |v ∈ λ}.

24 n n With the previous homogeneous coordinates in CP , the transition functions of γ1 in Ui ∩ Uj are deﬁned by xi/xj.

3. the “hyperplane” bundle H over CPn, dual of the canonical bundle. It is denoted by O(1) in algebraic geometry. With the previous homogeneous coordinates, the −1 transition functions of the hyperplane bundle in Ui ∩ Uj are defined by (xi/xj) . n−1 The hyperplane x0 = 0 with the induced orientation, is CP , that is a 2(n − 1)- n cycle in H2(n−1)(CP ; Z). The Poincarédual cohomology class hn is a generator of 2 n H (CP ; Z). We will see that c(H) = 1 + hn. 4. the universal bundle n n γr → Gr(C ) n is the set of all pairs {(P, v)} where P is an element of Gr(C ) and v a vector of P . n That is a vector bundle of rank r over Gr(C ). Every complex vector bundle ξ with rank r over a (paracompact) topological space ∗ n n X is isomorphic to f (γr ) for some f : X → Gr(C ) with sufficiently large n. 5. the universal bundle ∞ γr → Gr(C ) ∞ is the set of all pairs {(P, v)} where P is an element of Gr(C ) and v a vector of ∞ P . In particular, one has the bundle γ1 → CP .

The bundle γr is universal for all rank r-vector bundles.

n n 6. One deﬁnes the Stiefel manifold Vr(C ) which is the set of r-frames in C , that is n the set of ordered r-uples (v1, . . . , vr) of C-linearly independent vectors in C (see 0 Steenrod [Ste] where the Stiefel manifold is denoted by Wr,n). One has a homotopy

n ∼ Vr(C ) = Wr,n = U(n)/U(n − r).

n The fibre bundle Wr,n → Gr(C ) is a principal bundle with fibre and structural group U(r). n The fibre bundle Wr,n → Gr(C ) is an universal bundle for bundles which basis has dimension ≤ 2(n − r). n n n The vector bundle γr → Gr(C ) is a bundle associated to Wr,n → Gr(C ) with fibre Cr. 7. One define the bundle of complex r-frames tangent to the complex n-manifold M, i.e. the set of all pairs (x, (v1, . . . , vr)) where x is a point of M and (v1, . . . , vr) is a r-frame in the fibre TxM over x. That is the fibre bundle whose fibre at x is the manifold Vr(TxM) consisting of all complex r-frames in TxM. The “typical” fibre n is the Stiefel manifold Vr(C ). Note that a section of this bundle in complex Stiefel manifolds is a r-uple of C- linearly independent sections of the complex vector bundle TM.

25 3.2 General obstruction theory Let us recall the idea of the construction of characteristic classes by obstruction theory, following Steenrod [Ste], part III. We have seen that the meaning of Poincaré-HopfTheorem is that the Euler-Poincaré characteristic of a manifold M is a measure of the obstruction for the construction of a vector field tangent to M. In a more general way, the aim of the obstruction theory is to define invariants providing a measure of the obstruction to the construction of linearly independent sections of vector bundles. In a more precise way, the objective is to answer to questions of the following type: Let E be a vector bundle of rank n on a variety X and fix r such that 1 ≤ r ≤ n, is it possible to construct r sections of E, linearly independent everywhere? It is obviously possible to define such sections on the 0-skeleton of a triangulation of X. So, the question becomes the following: Let us consider a triangulation of X. Performing the construction of r independent sections by increasing dimension of the simplexes, up to what dimension can we proceed? Arriving to this obstruction dimension, is it possible to evaluate the obstruction? At that point, let us make a comment: Classical obstruction theory uses a triangulation of the considered space. In the following we will use a slightly different viewpoint, taking into account the fact that we want to deal with the singular case. It appears that in the singular case, the good decomposition to be taken into account for the construction of the sections is not a triangulation of the space but a dual cell decomposition in the ambient space. That is the reason for which, we will work on a cell decomposition, already in the non-singular case.

Let us consider a (simplicial or cellular) complex K and a subcomplex L. We will denote by X = |K| and Y = |L| the respective geometric realisations. The q-skeleton of K is denoted by Kq, that is the subcomplex consisting of all simplexes (or cells) whose q dimension is less or equal to q. Let us denote Xq = |K | the associated space. We consider a fibre bundle E with basis X and fibre F . To consider a section defined in a trivialisation open subset for the bundle provides a map with target F , as we already seen, see for instance (3.16). Aim of obstruction theory is to describe the problem of extension of maps f : Y → F to all of X, by successive extensions of the map from Xq to Xq+1. Let us suppose that p the function f : X → F is already known on Xp−1 and let us denote it by fp−1. Let d p an oriented p-cell, fp−1 is well defined on the boundary ∂d and determines an element [fp−1|∂dp ] ∈ πp−1(F ).

p Deﬁnition 3.5 The relative cochain denoted by c(fp−1) ∈ C (K,L; πp−1(F )) and deﬁned by p c(fp−1)(d ) = [fp−1|∂dp ] ∈ πp−1(F ) (3.6)

is called obstruction cochain (for the extension of fp−1 to Xp).

The function fp−1 can be extended to Xp if and only if c(fp−1) = 0. In particular, if πi(F ) = 0 for i = 1, . . . , j − 1, then every function fY : Y → F can be extended to fj : Xj → F .

26 Lemma 3.7 If fp−1 is homotopic to gp−1, then c(fp−1) = c(gp−1). ∼ Proof: In fact, as fp−1|∂dp = gp−1|∂dp , one has [fp−1|∂dp ] = [gp−1|∂dp ].

Theorem 3.8 c(fp−1) is a cocycle.

p+1 p+1 Proof: Let τ a (p + 1)-cell. One has to show that δ[c(fp−1)](τ ) = 0. One has

p+1 p+1 X p+1 p p δ[c(fp−1)](τ ) = c(fp−1)[∂τ ] = c(fp−1)( [τ : di ]di ) =

X p+1 p p X p+1 p [τ : d ]c(f )(d ) = [τ : d ][f | p ] i p−1 i i p−1 ∂di p p+1 where the sum is taken on all cells di which are faces of τ . Let us suppose that incidence p p+1 p+1 of all faces d of τ with τ is positive, then denoting f | p = α , one has P α = 0, i p−1 ∂di i i p p+1 p and the result. If incidence is not positive, then [τ : di ][fp−1|∂d ] = αi is the element i p of πp−1(F ) obtained from the function fp−1 restricted to the boundary of the face di with p+1 p+1 p the orientation induced from τ and one has P[τ : d ][f | p ] = P α = 0. i p−1 ∂di i

3.2.1 The diﬀerence cochain

Let fp−1 and gp−1 two extensions on Xp−1 of the same fp−2 : Xp−2 → F . We intend to provide a “measure” of their difference. Let dp−1 a (p − 1)-cell in (K). The two functions p−1 p−1 fp−1|dp−1 and gp−1|dp−1 coincide on the boundary ∂d . The cell d is homeomorphic p−1 p−1 both to the north hemisphere D+ , and to the south hemisphere D− , of the sphere p−1 p−1 p−1 S . One can interpret fp−1|dp−1 , resp. gp−1|dp−1 , as a function of D+ , resp. D− in F . These functions coincide on the equator Sp−2, homeomorphic to ∂dp−1, hence they define p−1 a function γ : S → F . Annulation of the homotopy class [γ] ∈ πp−1(F ) is a necessary and sufficient condition to deform gp−1|dp−1 in fp−1|dp−1 .

p−1 Definition 3.9 The difference cochain d(fp−1, gp−1) ∈ C (K,L; πp−1(F )) is defined by

p−1 p d(fp−1, gp−1)(d ) = (−1) [γ] ∈ πp−1(F ).

The diﬀerence cochain is a relative cochain of K modulo L. It vanishes if and only if ∼ fp−1 = gp−1 relatively to Xp−2. If hp−1 is a third extension of fp−2 then one has

d(fp−1, hp−1) = d(fp−1, gp−1) + d(gp−1, hp−1).

p−1 p−1 If fp−1 is an extension of fp−2 and c ∈ C (K,L; πp−1(F )) is a relative cochain, then p−1 there is an extension gp−1 of fp−2 such that d(fp−1, gp−1) = c .

Theorem 3.10 One has

δd(fp−1, gp−1) = c(fp−1) − c(gp−1).

That means that the diﬀerence of the obstruction cocycles of two extensions of fp−2 is a coboundary.

27 Lemma 3.11 If fp−2 can be extended as a function fp−1 : Xp−1 → F , then all obstruction cocycles c(fp−1) for extension of fp−2 to Xp−1 belong to the same cohomology class

p c¯(f) ∈ H (K,L; πp−1(F )).

Theorem 3.12 Let fp−1 : Xp−1 → F , then fp−2 extends to fp : Xp → F if and only if c¯(f) = 0.

3.2.2 The obstruction class

p Here we are using cohomology with local coeﬃcients H (K; {πp−1(F )}), i.e. bundle of abelian groups which associate to each point x of X the coeﬃcient group πp−1(Fx). Let us suppose that πi(F ) = 0 for i ≤ p − 2. Then one can construct a fonction fi without singularity for 1 ≤ i ≤ p − 1.

Deﬁnition 3.13 Let us suppose that πi(F ) = 0 for i ≤ p − 2. The primary obstruction class is the class of the obstruction cocycle [c(fp−1)], that is

p c¯(f) ∈ H (K,L; πp−1(F )).

Let us remark that in general, the system of coeﬃcients {πp−1(F )} is twisted.

3.3 Case of the tangent bundle We study the particular case of the (real) tangent bundle to a differentiable smooth manifold or the (complex) tangent bundle to an analytic complex manifold. We will denote by K the field R or C, according to the situation. Let M be a manifold of dimension n, over K, endowed with an euclidean (or hermitian) metric. The tangent bundle to M, denoted by TM, is a vector bundle of rank n over K, whose fibre in a point x of M is the tangent vector space to M in x, denoted by Tx(M) and is isomorphic to Kn. The vector bundle TM is locally trivial, i.e. there is a covering of M by open subsets such that the restriction of TM to U is isomorphic to U × Kn. The objective is to evaluate the obstruction to the construction of r sections of TM linearly independent (over K) in each point, i.e. an r-frame:

(r) Definition 3.14 An r-field on a subset A of M is a set v = {v1, . . . , vr} of r continuous vector fields tangent to M, defined on A. A singular point of v(r) is a point where the vectors (vi) fail to be linearly independent. A non-singular r-field is called an r-frame.

The r-frames are sections of the fibre bundle Vr(TM) over M. That is the fibre bundle associated to TM and whose fibre at the point x of M is the set of r-frames of Tx(M). n The fibre is the Stiefel manifold denoted by Vr(K ) that we described in 3.1.3 in the real case and in 3.1.4 in the complex case. To construct r linearly independent sections of TM over a subset A of M is equivalent to construct a section of Vr(TM) over A. Let us consider the following situation: (K) is a cell decomposition of M sufficiently small so that every cell d is included in an open subset U over which Vr(TM) is trivial.

28 One remarks that trivialisation open sets for Vr(TM) are the same that the ones of T (M). There exists always such a cell decomposition. Let us consider the following question: (r) Let us suppose that one has a section v of Vr(TM) on the boundary ∂d of the k-cell d. Is it possible to extend this section in the interior of d ? Is the answer is no, can one evaluate the obstruction for such an extension ? In order to answer the question, we need to define the notion of index of an r-field in a singular point and we need some notions and results on general obstruction theory. That is aim of the following sections. Then we will apply these results to the real and complex case, that is to define Stiefel-Whitney and Chern classes.

3.3.1 Index of a r-frame Let us consider an r-field v(r) defined on the boundary ∂d of a k-cell of the cell decom- (r) position (D) of M. In the same way than in 2.2, v is a section of the bundle Vr(TM), defined on the boundary of d. It provides a map

(r) v ∼ n pr2 n ∂d −→ Vr(TM)|U = U × Vr(K ) −→ V(r)(K ), (3.15)

where pr2 is the second projection. One obtains a map (r) k−1 ∼ pr2◦v n S = ∂d −→ Vr(K ) (3.16) n (r) which deﬁnes an element of πk−1(V(r)(K )) denoted by [ξ(v , d)]. Let us suppose that [ξ(v(r), d)] = 0, then, by classical homotopy theory, the map k−1 n k−1 k S → Vr(K ) deﬁned on the boundary S of the ball B can be extended inside the (r) n ball. In another words, if [ξ(v , d)] = 0, then the map ∂d → Vr(K ), i.e. the r-frame, can be extended inside the cell d. One can resume the situation by the following diagram similar to (2.7).

∼ k−1 n ∂d = S −−→ Vr(K )    ? % y ∼ k d = B This means that there is no obstruction to the extension of the section v(r) inside d. This n happens for example in the case πk−1(Vr(K )) = 0. In order to answer to the previous question, we need to know the homotopy groups n n of Vr(K ). The homotopy groups πk−1(Vr(K )) have been computed by Stiefel and by Whitney (see [Sti]) in the cases K = R and C. One has the following result: n n Let Vr(R ) be the Stiefel manifold of r-frames in R , one has:  0 for i < n − r n  πi(Vr(R )) = Z for i = n − r even or i = n − 1 if r = 1 (3.17)  Z2 for i = n − r odd and r > 1

29 n n For the Stiefel manifold Vr(C ) of (complex) r-frames in C , one has: ( n 0 for i < 2n − 2r + 1 πi(Vr(C )) = (3.18) Z for i = 2n − 2r + 1

A generator of the first non-zero homotopy group can be described in the following way. Let us give it in the real framework, then in the complex one. n In the real case, let us denote p = n−r +1, one describes a generator of πn−r(Vr(R )). Let us fix a (r − 1)-frame in Rn. It defines a (r − 1)-subspace of Rn whose complementary is a real space Rp. The unit sphere in Rp denoted by Sp−1 is oriented. Let us consider, for each point w of the sphere, a r-frame consisting of the vector w and the fixed (r − 1)- n p−1 frame, one obtains an element of Vr(R ). The induced map from the oriented sphere S n n to Vr(R ) defines a generator of πp−1(Vr(R )). In the complex case, let us denote 2p = 2(n − r + 1), one describes a generator of n ∼ n n π2p−1(Vr(C )) = Z. Let us fix a (r − 1)-frame in C . It defines a (r − 1)-subspace of C whose complementary is a complex space Cp. The unit sphere in Cp denoted by S2p−1 is oriented, the orientation being induced by the natural one of Cp. Let us consider, for each point w of the sphere, a r-frame consisting of the vector w and the fixed (r − 1)-frame, n 2p−1 one obtains an element of Vr(C ). The induced map from the oriented sphere S to n n Vr(C ) defines a generator of π2p−1(Vr(C )). One obtain the following result:

Proposition 3.19 – Real case. Let us consider an r-frame v(r) deﬁned on the boundary of the k-cell d.

• If k < n − r + 1, one has [ξ(v(r), d)] = 0 and one can extend the r-frame deﬁned on ∂d inside d without singularity.

• If r = 1 and k = n, one can extend the vector field v(1) = v defined on ∂d inside d with an isolated singularity in the barycenter dˆ of the cell, with index [ξ(v, d)] = I(v, dˆ). That is the index we defined in Definition 2.1.

• If r > 1 and k = n − r + 1, then one can extend the r-frame v(r) deﬁned on ∂d inside d with an isolated singularity at the barycenter dˆ of the cell. In that case, [ξ(v(r), d)] is an integer if k is odd and an integer mod 2 if k is even. Reducing modulo 2, one obtains an index I(v(r), dˆ) that measures the obstruction to the extension of v(r) inside the k-cell d.

The dimension p = n − r + 1 is called the obstruction dimension for the construction of a r-frame tangent to M.

Proposition 3.20 – Complex case. Let us consider a complex r-frame v(r) deﬁned on the boundary ∂d of the k-cell d.

• If k < 2(n − r + 1), one has [ξ(v(r), d)] = 0 and one can extend the r-frame v(r) inside d without singularity.

30 • If k = 2(n−r+1), then one can extend the r-frame v(r) inside d with an isolated singularity at the barycenter dôf the cell. In that case, one obtain an index [ξ(v(r), d)] ∈ Z that we define as I(v(r), dˆ). The index measures the obstruction to the extension of v(r), defined on the boundary ∂d, inside d. The dimension 2p = 2(n − r + 1) is called the obstruction dimension for the construction of a complex r-frame tangent to M.

3.4 Applications: Stiefel-Whitney and Chern classes Let us apply the previous construction to the cases of r-ﬁelds tangent to a manifold, in the real and the complex case.

3.4.1 Stiefel-Whitney classes The Stiefel-Whitney classes have been defined by obstruction theory (see [Sti],[Wh1]). In fact, Whitney used the same strategy than Stiefel, applying it to arbitrary sphere bundles. We use the Steenrod construction ([Ste], part III). The pth Stiefel-Whitney class of M, denoted by wp(M), is defined as the primary obstruction to constructing an r-frame over M, that is a section of Vr(TM) or a set of r linearly independent vector fields tangent to M, with p = n − r + 1. More precisely, one performs the following construction: Using the result in (3.17) one can construct an r-frame by choosing any r-frame v(r) on the 0-skeleton of the cell decomposition (D), then extending it without zeroes till the obstruction dimension p = n − r + 1. That means that v(r) has no singularity on the (p − 1)-skeleton and isolated singularities on the p-skeleton of (D). Given the r-frame v(r) on the boundary of each p-cell d, one can extend v(r) on d with a singularity at the barycenter dˆ of index ( (r) ˆ (r) n Z for p = n − r + 1 odd or p = n if r = 1 I(v , d) = [(v )p−1|∂dp ] ∈ πp−1(Vr(R )) = Z2 for p = n − r + 1 even and r > 1, using the notation in 3.6. In any case, there is a non trivial homomorphism from πp−1(F ) (r) ˆ to Z2. hence we can reduce the coefficients modulo 2 obtaining I(v , d) ∈ Z2. P (r) ˆ ∗ p One can define the p-cochain I(v , d) d in C (D, Z2), its value on each p-cell d is I(v(r), dˆ). According to Theorem 3.8, the cochain is in fact a cocycle and defines an p p element w (M) in H (M; Z2). The Definition 3.13 provides the following:

p p Definition 3.21 The p-th Stiefel-Whitney class of M, denoted by w (M) ∈ H (M; Z2) is the class of the primary obstruction cocycle corresponding to constructing an r-frame tangent to M. By the general obstruction theory, the obtained classes do not depend on the choices we make in the construction. In the particular case r = 1, one can use integer coefficients. The evaluation of wm(M) ∈ Hm(M; Z) on the fundamental class [M] of M is the Euler-Poincarécharacter- istic of M.

31 Let us suppose that the cell decomposition (D) is obtained by duality of a triangulation (K) of M. Each p-cell d = d(σ) in (D) is dual of an (r −1)-simplex σ in (K). By Poincar´e duality (cap-product by the fundamental class),

m−r+1 H (M; Z2) −→ Hr−1(M; Z2) the image of d∗ is σ and image of wp(M) is the so-called (r − 1)-homology Siefel-Whitney class, denoted by wr−1(M). A cycle representing wr−1(M) is given (mod 2) by X I(v(r), dˆ(σ))σ. dim σ=r−1

In fact, Stiefel-Whitney classes can be deﬁned in a combinatorial way [HT].

3.4.2 Chern classes The definition of Chern classes by obstruction theory in the complex case is similar to the real case, even simpler. Let M denote an analytic complex manifold and TM the complex tangent bundle to M. The pth Chern class of M, denoted by cp(M), will be defined as the primary obstruction to constructing a complex r-frame over M, that is a section of Vr(TM) or a set of r linearly independent vector fields tangent to M, with p = n − r + 1. Using the result in (3.18) one can construct an r-frame by choosing any r-frame v(r) on the 0-skeleton of the cell decomposition (D), then extending it without zeroes till the obstruction dimension 2p = 2(n − r + 1). (3.22) That means that v(r) has no singularity on the (2p−1)-skeleton and isolated singularities on the 2p-skeleton of (D). Given the r-frame v(r) on the boundary of each 2p-cell d, one can extend v(r) on d with a singularity at the barycenter dˆ of index

(r) ˆ (r) n I(v , d) = [(v )2p−1|∂d2p ] ∈ π2p−1(Vr(C )) = Z using the notation in 3.6. n The generators of π2p−1(Vr(C )) being consistent (see [Ste]), one can define the 2p- P (r) ∗ 2p (r) cochain I(v , dˆ) d in C (D, Z), its value on each 2p-cell d is I(v , dˆ). this defines a cochain 2q γ ∈ C (M; π2q−1(Wr,m)) , by γ(d) = I(v(r), d), for each 2q-cell d, and then extend it by linearity. One can define the 2p-cochain

X (r) ∗ 2p I(v , dˆ) d ∈ C (K, Z) (3.22) whose value on each 2p-cell d is I(v(r), dˆ). According to Theorem 3.8, the cochain is in fact a cocycle and deﬁnes an element cp(M) in H2p(M; Z). The Deﬁnition 3.13 provides the following:

32 Deﬁnition 3.23 The p-th Chern class of M, denoted by cp(M) ∈ H2p(M; Z) is the class of the primary obstruction cocycle corresponding to constructing a complex r-frame tangent to M.

By the general obstruction theory, the obtained classes do not depend on the choices we make in the construction. In the particular case r = 1, the evaluation of cm(M) on the fundamental class [M] of M yields the Euler-Poincar´echaracteristic of M. Note that cm(M) coincides with the Euler class of the underlying real tangent bundle

TRM, so these classes are natural generalization of the Euler class. Let us suppose that the cell decomposition (D) is obtained by duality of a triangulation (K) of M. Each 2p-cell d = d(σ) in (D) is dual of an 2(r − 1)-simplex σ in (K). By Poincar´eduality (cap-product by the fundamental class),

2(m−r+1) H (M; Z) −→ H2(r−1)(M; Z) the image of d∗ is σ and image of cp(M) is the so-called 2(r − 1)-homology Chern class, denoted by cr−1(M). A cycle representing cr−1(M) is given by X I(v(r), dˆ(σ))σ. (3.23) dim σ=2(r−1) As we will see in the next sections, there is no cohomology Chern class in the case of singular varieties, but expression (3.23) will generalise for suitable frames we will deﬁne.

3.5 Axiomatic deﬁnition Deﬁnition 3.24 [Theorem] To each complex vector bundle ξ of rank n over a space M, 2i one can associate a class c(ξ) = 1 + c1(ξ) + ··· + cn(ξ), where ci(ξ) ∈ H (M; Z) and ci(ξ) = 0 if i > n, satisfying the following properties: 1. (Naturality) For each f : Y → M, then f ∗(c(ξ)) = c(f ∗(ξ)). 2. (Whitney sum) If ξ and η are two bundles over M, then c(ξ ⊕ η) = c(ξ) ∪ c(η).

1 2 1 1 3. the class c1(γ1 ) ∈ H (CP ; Z) of the canonical line bundle over CP (see 3.4) is non zero.

Modulo naturality, the last axiom is equivalent to the following ones:

0 n n n 4 . Let γ1 be the canonical bundle over CP , then c1(γ1 ) = hn is a generator of H2(CPn; Z) (see 3.4). 00 ∞ 4 . Let γ1 be the canonical line bundle over CP , then c1(γ1) is a generator of the polynomial ring H∗(CP∞; Z).

Proposition 3.25 If ξ is a trivial bundle, then ci(ξ) = 0 for i ≥ 1.

33 4 Hirzebruch theory

In this section, one explicits the Hirzebruch theory that provides a way to unify, in the case of manifolds, the three theories of characteristic classes: the Chern class, the Todd class and the Thom-Hirzebruch class.

4.1 The arithmetic genus

Let gi be the number of C-linearly independent holomorphic diﬀerential i-forms on the n-dimensional complex algebraic manifold X.

• g0 is the number of linearly independent holomorphic functions, i.e. the number of connected components of X,

• gn is called geometric genus of X,

• g1 is called irregularity of X,

Deﬁnition 4.1 [Arithmetic Genus] The arithmetic genus of X, denoted by χa(X) is deﬁned as : n X i χa(X) := (−1) gi i=0

Example 4.2 Let X be a complex algebraic curve, i.e. a compact Riemann surface. X is homeomorphic to a sphere with g handles. Then g0 = 1 and g1 = gn = g. The arithmetic genus of X is:

χa(X) = 1 − g

4.2 The Todd genus The Todd genus T (X) has been deﬁned (by Todd) in terms of Eger-Todd fundamental classes (polar varieties), using Severi results. The Eger-Todd classes are homological Chern classes of X. Todd “proved” that

T (X) = χa(X). In fact, the Todd proof uses a Severi Lemma which has never been completely proved. The Todd result has been proved by Hirzebruch, using other methods.

34 4.3 The signature Deﬁnition 4.3 [Thom-Hirzebruch] Let M be a (real) compact oriented 4k-dimensional manifold. Then the map

2k 2k H (M; R) × H (M; R) −→ R, (x, y) 7→ hx ∪ y, [M]i ∈ R deﬁnes a bilinear form on the vector space H2k(M; R). The index (or signature) of M, denoted by sign(M), is deﬁned as the index of this form, i.e. the number of positive eigenvalues minus the number of negative eigenvalues.

4.4 Hirzebruch Theory The Hirzebruch theory uses multiplicative series and Chern root. let us deﬁne (or recall) these two ingredients.

4.4.1 Hirzebruch Series

For y ∈ R, let us deﬁne the Hirzebruch multiplicative series (for a review on multiplicative series, see Hirzebruch [Hirz]):

α(1 + y) Qy(α) := − αy ∈ Q[y][[α]] 1 − e−α(1+y) One has the particular cases:

• Q−1(α) = 1 + α y = −1

• Q (α) = α y = 0 0 1 − e−α

α • Q1(α) = tanh α y = 1

4.5 Characteristic Classes of Manifolds

Let X be a complex manifold with dimension dimC X = n, let us denote by

n ∗ X j j 2j c (TX) = c (TX), c (TX) ∈ H (X; Z) j=0 the total Chern class of the (complex) tangent bundle TX.

2 Deﬁnition 4.4 The Chern roots αi of the complex manifold X are elements in H (X; Z) deﬁned by: n n X j j Y c (TX) t = (1 + αit). j=0 i=1

35 n Y For each y ∈ R, one deﬁnes the Todd-Hirzebruch class: tdg(y)(TX) := Qy(αi) i=1  n ∗ Y c (TX) = (1 + αi) y = −1   i=1   Chern class,     n  ∗ Y αi td (TX) = ( 1−e−αi ) y = 0 tdg(y)(TX) = i=1   Todd class,    n  Y L∗(TX) = ( αi ) y = 1  tanhαi  i=1   Thom-Hirzebruch L-class.

Remark 4.5 The previous equalities can be considered as deﬁnition of the Todd class and the Thom-Hirzebruch L-class for the reader who is not already aware of these notions.

4.6 The χy-characteristic Let X be a complex projective manifold.

Deﬁnition 4.6 For each y ∈ R, one deﬁnes the χy-characteristic of X by

<∞ <∞ ! X X i i p ∗ p χy(X) := (−1) dimC H (X, ∧ T X · y p=0 i=0

One has the particular cases:

• y = −1 χ−1(X) = χ(X), Euler-Poincar´echaracteristic of X (by Hodge theory),

• y = 0 χ0(X) = χa(X), arithmetic genus of X (by deﬁnition)

• y = 1 χ1(X) = sign(X), signature of X (by Hodge theory)

One has the following table of invariants:

y χy(X) tdg(y)(TX) -1 χ(X) c∗(TX) ∗ 0 χa(X) td (TX) 1 sign(X) L∗(TX)

36 4.7 Hirzebruch Riemann-Roch Theorem Theorem 4.7 (Hirzebruch Riemann-Roch Theorem) One has: Z χy(X) = tdg(y)(TX) ∩ [X] ∈ Q[y]. X In the three particular cases, one obtains:

R ∗ • χ(X) = X c (TX) ∩ [X] y = −1 Euler - Poincar´echaracteristic of X Poincar´e-HopfTheorem

R ∗ • χa(X) = X td (TX) ∩ [X] y = 0 arithmetic genus of X Hirzebruch-Riemann-Roch Theorem

R ∗ • sign(X) = X L (TX) ∩ [X] y = 1 signature of X Hirzebruch signature Theorem

37 5 Singular varieties

A singular variety is a variety which contains singular points, that is points for which the property in Definition 1.12 is not satisfied. Examples of singular varieties are the following: The pinched torus: the pinched point a does not admit any neighbourhood satisfying the property 1.12. In that case, the link of an “elementary neighbourhood” of a is the union of two not connected circles. Another example is provided by the suspension of the torus. The two points a and b of the suspension of the torus are singular points, in that case, the link of a (or b) is a torus, it is not a sphere. In order to extend the notion of characteristic classes to singular varieties, it is necessary to know the local structure of the singular variety. That is given by the structure of stratified space and by suitable definition of triangulation on the variety.

5.1 Stratifications Definition 5.1 Let X be a topological space, we denote by X a filtration of X by closed subsets

∅ = X−1 ⊂ X0 ⊂ X1 ⊂ · · · ⊂ Xn−2 ⊂ Xn−1 ⊂ X = Xn (5.2) A topological stratification of X is the data of a filtration X of X such that each difference Vk = Xk − Xk−1 is either empty or a topological manifold of pure dimension k. The connected components of the Vk are called the strata. The stratifications that we will consider will be locally finite partitions of X into locally closed submanifolds, the strata, satisfying the frontier condition:

Vk ∩ V j 6= ∅ ⇒ Vk ⊂ V j

Let X be a closed subset of a differentiable manifold M.A differentiable stratification of X is a topological stratification X of X such that each stratum in Vk is a differentiable submanifold of M.

In order to work with, the considered stratification should satisfy conditions which precise the way the strata are glued together. On one hand, there are many ways to define these conditions, according to the specific problem. On the other hand, given conditions on the stratification, one has to know what kind of singular variety admits a stratification satisfying these conditions. In the following, one considers the stratifications which will be useful for the construction of characteristic classes. One refer to [B3, Tr] for more information on the different types of stratifications.

38 5.1.1 Whitney stratifications As we have seen, on a singular variety, there is no longer tangent space in the singular points. One way to find a substitute for the tangent bundle is to stratify the singular variety in submanifolds. One can proceed the following construction: If X is a singular analytic variety, equipped with a stratification and embedded in a smooth analytic manifold M, one can consider the union of tangent bundles to the strata, that is a subspace E of the tangent bundle to M. The space E is not a bundle but it generalizes the notion of tangent bundle in the following sense: A section of E over X is a section v of TM|X such that at each point x ∈ X the vector v(x) belongs to the tangent space of the stratum containing x. Such a section is called a stratified vector field over X. To consider E as the substitute for the tangent bundle of X and to use obstruction theory is the M.-H. Schwartz point of view (1965, [Sc2]) in the case of analytic varieties. When one consider stratification of singular varieties, it is natural to ask for conditions with which the strata glue together. The so-called Whitney conditions are those which allow one to proceed to the construction of radial extension of vector fields. According to a result of Whitney, every analytic variety can be equipped with a Whitney stratification. Definition 5.3 One says that the Whitney conditions are satisfied for a stratification if, for any pair of strata (Vi,Vj) such that Vi is in the closure of Vj, one has: a) if (xn) is a sequence of points in Vj with limit y ∈ Vi and if the sequence of tangent spaces Txn (Vj) admits a limit T (in the suitable grassmanian space) when n goes to +∞, then Ty(Vi) is included in T . b) if (xn) is a sequence of points in Vj with limit y ∈ Vi and if (yn) is a sequence of points in Vi with limit y, such that the sequence of tangent spaces Txn (Vj) admits a limit T for n going to +∞ and such that the sequence of directions xn yn admits a limit λ when n goes to +∞, then λ lies in T . Example The conditions (a) and (b) are not satisfied for the stratification of the cone X consisting of a generatix D = Vi and Vj = X \ D. This is clear taking for (xn) a sequence of points going to the vertex y of the cone, along a generatrix (different from D), and for yn a sequence of points such that the segment xnyn has always the same direction. Adding the vertex of the cone as a supplementary 0-dimensional stratum, the new stratification satisfies the Whitney conditions. Example Let V be the variety whose equation in C3 is y2 + x3 − t2x2 = 0, stratified by the horizontal axis Y = Vi and Vj = V − Vi, then the (a)-condition is satisfied but not (b). Adding the vertex of Cn as a new stratum, the Whitney conditions are verified.

5.2 Poincaréhomomorphism Let us recall (see [B1]) that the Poincaréhomomorphism can be described in the following way. Let us suppose that the oriented singular n-dimensional variety X is a subvariety of a P.L. oriented m-dimensional manifold M. Any stratification of X defines a stratification of M adding M − X as regular stratum. Let us denote by (K) a locally finite triangulation of M compatible with the stratification and by (K0) a barycentric subdivision of (K). The chain (or cochain) complexes relatively to (K) or (K0) will be denoted (K0) ∗ by C∗ (X),C(K)(X) for example.

39 0 Providing to all n-dimensional simplexes of (K ) the orientation of Xreg, the sum of (K0) these simplexes is a cycle in Cn (X). Its class in Hn(X) is the fundamental class of X, denoted by [X]. For every (n − i)-simplex σ in (K), the dual cell of σ in M, denoted by d(σ) has dimension m − (n − i). It is transverse to X, i.e. to every stratum Xn−α − Xn−α−1 of X. The intersection d(σ) ∩ X is an oriented i-dimensional (K0)-chain in X. Let us deﬁne the Poincar´ehomomorphism

n−i H (X) → Hi(X) given by the chain map n−i (K0) C(K) (X) → Ci (X) which maps the elementary (n − i)-cochain d∗ = d∗(σ), dual of the simplex σ in K, to the i-chain ξ = d(σ) ∩ X of K0. In other words, ξ is the cap-product of σ∗ by [X] (see [B1]).

5.2.1 Alexander isomorphism

m−i The Alexander isomorphism (see [B1]) H (M,M − X) → Hi(X) is induced by the isomorphism: m−i ˚ (K) C(D) (M,M − T ) → Ci (X) m−i ∗ m−i which associates to a D-cochain (d ) such that d ∩ X 6= ∅ the K-chain σi such that m−i d = d(σi).

5.3 Poincaré-HopfTheorem: The singular case If X is a singular variety, the Poincaré-HopfTheorem fails to be true. The main reason is that there is no longer tangent space at each singularity. The definition of the index of a vector field in one of its singular points takes sense on a smooth m-manifold only. In particular the singular point must have a neighbourhood isomorphic to the ball Bm and whose boundary is isomorphic to the sphere Sm−1. Let us consider the example of the pinched torus X in R3. The pinched point a is a singular point of X, in fact it constitutes the singular part of the pinched torus. The only ‘natural’ way to define an index of a vector field at the point a is to consider a vector field v defined in a ball B3(a) centered in a, in R3, with an isolated singularity at a, such that if x ∈ X \{a}, then v(x) is tangent to the smooth manifold X \{a} and such that v does not have other singularities in B3(a). Let us consider two examples of such vector fields: a) The vector field tangent to the parallels of the torus T determines, on the pinched torus X, a vector field v pointing inward the ball B3(a) along one of the two unlinked circles, which are the intersection of ∂B3(a) and X, and pointing outward the ball along the other circle. On the one hand, this vector field, defined on ∂B3(a)∩X, is the restriction of a vector field w defined on ∂B3(a) with index 0 at a. On the other hand, there is no more singularity of v on X \{a}. In this case, the Poincaré-HopfTheorem is not satisfied: one has χ(X) = 1 6= 0 = I(w, a).

40 b) Let us consider a vector field v pointing outward the ball B3(a) along ∂B3(a) and tangent to X along the restriction ∂B3(a) ∩ X. This vector field has index +1 at a. It is orthogonal to the two meridians ∂B3(a) ∩ X and it can be extended on the pinched torus as a continuous vector field without other singularity. In fact, one can define an extension v such that, on each meridian, the angle of v(x) with the tangent space to the meridian is constant and this angle goes down continuously as the distance to a grows till being 0 for the meridian opposed to a. In this case, the Poincaré-HopfTheorem is valid: χ(X) = 1 = I(v, a). The vector field v is the first example of M.-H. Schwartz’s radial vector field, of which we will make a systematic study. If X is a singular variety, the Poincaré-HopfTheorem fails to be true, the main reason is that there is no more tangent space in singular points. The definition of the index of a vector field at one of its singular points takes sense on a smooth manifold only, the reason being that the link of a point is a sphere. In order to obtain a Poincaré-HopfTheorem, one can think to consider a stratification of the singular variety (see section 5.1), i.e. a decomposition of the singular variety into smooth manifolds and consider continuous vector fields which are stratified. Definition 5.4 A stratified vector field v on X is a (continuous) section of the tangent bundle of M, T (M), such that, for every x ∈ X, then one has v(x) ∈ T (Vi(x)) where Vi(x) is the stratum containing x. Proposition 5.5 [Sc4] If the stratification satisfies Whitney conditions (a) and (b), then there exists on X a stratified vector field v with isolated singularities ak. The index I(v|V , ak), defined in the stratum Vi(x) containing x, is well defined. i(ak) One could define the index of a stratified vector field v at a singular point a situated in the stratum Vi as the index of the restriction I(v|Vi , a). The natural generalization of the Poincaré-HopfTheorem to singular varieties would be the following formula: X χ(X) = I(v|V , ak). (5.6) i(ak) ak∈Sing(v)

In general, the formula(5.6) is not true. Let us provide the (counter)-example given in [Sc4, 6.2.1]:

Example 5.7 In a first step, in R2 with coordinates (x, y), one considers the (closed) balls centered at the origin, B with radius 1 and D with radius 2. One has χ(D) = +1. Inside the ball B, one considers the continuous vector field v1(x, y) = (|x|, y). One has v1(0) = 0, the point 0 is a singularity of v1 with index I(v1, 0) = 0. On the boundary ∂D, one considers the vector field v2(x, y) = (x, y) pointing radially outwards. One can extend v2 inside D as a continuous vector field v which is v2 along ∂D, v1 inside B and which is tangent to the y-axis Y along Y . The vector field defined by ( 2|x| − x + (x − |x|)px2 + y2, y on D \ B v(x, y) = v1(x, y) = (|x|, y) inside B

41 satisfies the conditions. The vector field v has an isolated singular point of index 0 at 0 and another isolated singular point at a = (−3/2, 0) ∈ D\B. By Poincaré-HopfTheorem with boundary(2.4.3), one has χ(D) = +1 = I(v, 0) + I(v, a), that implies I(v, a) = +1. Let us remark that while I(v, 0) = 0, one has I(v|Y , 0) = +1. Now fold the picture along the y-axis, as a (differentiable) singular surface x2 − z3 = 0 in R3. In that case, D becomes a singular variety ∆, with boundary and stratified by Y and ∆\Y . The vector field v in D defines a stratified vector field, still denoted by v in ∆. It has two isolated singular points: 0 and a. One has I(v|Y , 0) = +1 and I(v, a) = +1. The formula (5.6) would be written:

χ(∆) = +1 6= I(v, a) + I(v|Y , 0) = 1 + 1 = 2.

Let us remark that the vector ﬁeld v is not “radial” at the singular point 0, it is not pointing outwards the unit ball centered at 0.

The main result of M.H. Schwartz is to provide, for singular varieties, an explicit construction of certain vector fields, called radial vector fields for which the Poincaré- Hopf theorem is still valid. Moreover, in the same way that the radial vector fields allow to recover the Poincaré- Hopf Theorem, the construction of characteristic classes for singular varieties will consist in a construction of vector frames adapted to the singular situation and generalising the notion of radial vector fields.

5.3.1 Radial extension process One gives a description of the local radial extension process. This will be used for the global process in the next section. The local radial extension, defined by M.-H. Schwartz, is similar to the one defined by Milnor in order to prove Theorem 2.11. Let X be a singular variety embedded in an m-dimensional manifold M. Let Vα ⊂ X be a stratum, B(a) ⊂ Vα a neighbourhood of the point a in Vα and v a vector field defined on B(a) with an isolated singularity at a. One can construct two vector fields: 1. Parallel Extension. We will denote by N(a) a tubular neighbourhood homeomorphic k k to B(a) × D , where D is a disk transverse to Vα and k = m − dim Vα. Let us consider the parallel extensionv ˆ of v in the tube N(a). Let Vβ be a stratum such that a ∈ Vβ. At a point x ∈ N(a) the parallel extensionv ˆ(x) is not necessarily tangent to Vβ. However, the Whitney condition (a) guarantees that if N(a) is sufficiently small, then the angle between Ta(Vα) and Tx(Vβ) is small. That implies that the orthogonal projection ofv ˆ(x) onto Tx(Vβ) does not vanish. Of course, considering for each stratum the projection of the parallel extension on the tangent space to the stratum at the given point does not provide a continuous vector field. In order to obtain a continuous vector field, one has

42 to consider a slight modification of the construction in the neighbourhood of the strata, which is easy to understand, but complicated to describe in details. A good extension will bev ˆ(x) away from Vβ and continuously going to the projection ofv ˆ(x) onto Tx(Vβ) when approaching Vβ, using a suitable partition of unity. The construction of such an extension is correctly and entirely described in M.-H. Schwartz’s book [Sc4]. In fact, one has to work simultaneously for all strata Vβ such that a ∈ Vβ, that complicates a detailed construction. In conclusion, the Whitney condition (a) implies that one can proceed to the construction of a stratified vector field, still denoted byv ˆ(x), which is a “parallel extension” of the given vector field on Vα, in a suitably small tubular neighbourhood around B(a) in M. One observes that the singular locus ofv ˆ corresponds to an (m − dim Vα)-dimensional disk which is transversal to B(a) in M. 2. Transversal vector field. Let us consider the transversal vector field τ(x), as in the proof of Theorem 2.11. This vector field is essentially the gradient of the function “square of the distance to Vα”, for an appropriate Riemannian metric. The vector field τ(x) is not necessarily tangent to the strata Vβ such that a ∈ Vβ. However, the Whitney condition (b) guarantees that in a sufficiently small “tube” around B(a), the angle between τ(x) and Tx(Vβ) is small. That means that the orthogonal projection of τ(x) onto Tx(Vβ) does not vanish. In the same way as for the parallel extension, for each stratum one could consider the projection of τ(x) onto the tangent space to the stratum at the given point. However, this does not provide a continuous vector field. In order to obtain a continuous vector field, one has to consider a similar modification of the construction. The correct vector field is τ(x) away from Vβ and continuously going to the projection of τ(x) onto Tx(Vβ) when approaching Vβ. That construction is also completely described in M.-H. Schwartz’s book [Sc4], and one has to work simultaneously for all strata Vβ such that a ∈ Vβ. ∼ k k k−1 In the boundary of the tube N(a) = B(a) × D the part B(a) × ∂D = B(a) × S shall be called the horizontal part. In conclusion, one obtains a stratified “transversal” vector field, still denoted by τ(x), which is zero along Vα and growing with the distance to Vα and which is pointing outward the horizontal part of the boundary of the tube N(a) provided that the tube is sufficiently small.

Definition 5.8 The radial extension of the vector field v defined on B(a) ⊂ Vα is the vector fieldv ˜ defined on the tube N(a) as the sum

v˜(x) =v ˆ(x) + τ(x).

Proposition 5.9 The radial extension v˜ of the vector ﬁeld v is transversal to the boundaries of the tube N(a) around B(a), pointing outward the horizontal part of ∂N(a). Its unique singularity inside N(a) is a, i.e. the same singularity as the initial vector ﬁeld v. Moreover, the index of v˜ in a, computed in the tube N(a), is the same as the index of v at a, computed in the manifold Vα, namely we have

I(˜v, a; M) = I(v, a; Vα).

43 This property, i.e., the above equality, is the main property of the radial extension, that is precisely the property which allows one to prove the Poincar´e-HopfTheorem for singular varieties.

5.3.2 Poincaré-HopfTheorem for singular varieties. In this section one proceeds to the construction of a “global” radial extension of a vector field, which M.-H. Schwartz called radial vector field and one shows the following:

Theorem 5.10 Let X ⊂ M be a (compact) singular variety embedded in a manifold M. One can construct on X a (stratified) radial vector field, in the sense of M.-H. Schwartz. That is a vector field v defined in a tubular neighbourhood Nε(X) of X in M, pointing outward Nε(X) along the boundary. It has finitely many isolated singularities ai in Nε(X), all situated in X, and one has

I(v, ai; M) = I(v|Vα(i) , ai; Vα(i)), where Vα(i) is the stratum of X containing ai.

Proof: The “global” construction of the radial vector field goes as follows: One consider a Whitney stratification on X as before and one adds the stratum M \X in order to obtain a Whitney stratification of M. The aim of the process is to construct, by increasing induction on the dimension of the strata of X, a stratified vector field v in a neighbourhood of X in M, with finitely many isolated singularities ai in the strata Vα, such that if ai ∈ Vα, the index of v at ai is the same, computed in Vα or in M. By the induction process on the dimension of the strata, we will show the following:

α (P) For each stratum Vα, there is a neighbourhood Nε around V α and a stratified α α vector field v defined on Nε , pointing outward Nε along the boundary, with isolated singularities ai in V α and such that if Vβ is the stratum containing ai, then the index α of ai computed in Vβ is the same as the index of ai computed in Nε , i.e. in M. Namely, we will denote

I(v, ai) = I(v, ai; Vβ) = I(v, ai; M).

α The neighbourhood Nε is the set of points in M with distance less than ε from points in V α. That is not a fibre bundle over V α, but the following construction shows that for each stratum Vβ ⊂ V α, there is a neighbourhood Aβ of V β \ Vβ in Vβ such that α the restriction of Nε to Aβ is a fibre bundle with fibre a disk whose dimension is the codimension of Vβ in M. Let us show that induction property (P) is true for the lowest dimensional stratum. If the lowest dimensional strata in X are 0-dimensional ones, i.e. V0 is a set of finitely many points ak, then one considers a radial vector field v in a ball Bε(ak) centered at each of these points. According to the Bertini-Sard Lemma, the boundary ∂Bε(ak) is transverse to the strata Vγ containing ak in their closure (one takes for ε the smallest of ε

44 for all ak). For each point x ∈ Vγ ∩ ∂Bε(ak), the radial vector field v(x) is not orthogonal to Tx(Vγ). One can deform the radial vector field to a stratified vector field v pointing outward Bε(ak) along ∂Bε(ak). In this case, the index I(v, ak) is +1 and obviously one has X χ(V0) = I(v, ak). k

0 In this case, Nε is the union of Bε(ak). If the lowest dimensional stratum is a stratum Vα of dimension s > 0, then one constructs, by classical obstruction theory, a vector field v on Vα with finitely many isolated singularities ai. We notice that Vα is a manifold without boundary and it has to be compact if X is compact. According to the classical Poincaré-HopfTheorem 2.12, one has X χ(Vα) = I(v, ai). i α The extension process in a neighbourhood Nε of Vα, described in the proof of Theorem 2.11, can be slightly deformed according to the local case (cf Proposition 5.9) in order to α α obtain a stratified vector field defined on Nε , pointing outward Nε along its boundary and such that the index of v at each singularity ai is the same, whether it is computed in α Vα or in Nε , i.e. in M. Let us now suppose that induction property (P) holds for all strata up to Vα (i.e. for all strata whose dimension is less than or equal to dim Vα) and let us call Vγ the following one. One has to show that (P) holds for Vγ. α The vector field v is defined on Uγ = Vγ ∩ Nε and is pointing inward Vγ along ∂Uγ = U γ \ Uγ. By classical obstruction theory, one can extend v inside Vγ, as a vector field still denoted by v, with finitely many isolated singularities aj in Vγ \ Uγ. α For t ∈]0, 1], let us denote by Ntε the (open) neighbourhood of V α, which is the set of α points whose distance to V α is less than tε. The vector field v defined on Vγ \ Nε/2 can 0 0 be extended, using the local extension process, as a vector field v defined in a tube Nη α around Vγ \ Nε/2 without other singularities than the points aj and such that

0 I(v, aj; Vγ) = I(v, aj; Nη).

0 α Let us notice that Nη is a disk bundle with the basis Vγ \ Nε/2 and the fiber a disk whose dimension is the codimension of Vγ in M. α 0 α 0 On the intersection Nε ∩Nη, one has two vector fields: v defined on Nε and v defined 0 0 on Nη. They coincide on Vγ. The vector field defined by w = (2 − 2t)v + (2t − 1)v on α α 0 0 each Ntε, for t ∈ [1/2, 1], coincides with v on ∂Nε/2 and with v on ∂Nη. α 0 0 α α 0 The vector field defined as v on Nε/2, v on Nη \ Nε and w on Nε ∩ Nη satisfies the γ α 0 property (P) for Vγ with the neighbourhood Nε defined as Nε/2 ∪ Nη. At the last step, one denotes by Nε(X) the neighbourhood of X constructed by the induction process. By construction, the stratified radial vector field v satisfies the statement of the theorem.

45 Theorem 5.11 (Poincaré-HopfTheorem for singular varieties) Let X ⊂ M be a (compact) singular variety embedded in a manifold M, and v be a stratified radial vector field defined in the neighbourhood Nε(X) of X (Theorem 5.10). Then one has X χ(X) = I(v, ai). i

Proof: Using the stratiﬁed radial vector ﬁeld constructed in the proof of Theorem 5.10 and Theorem 2.16, one has X χ(Nε(X)) = I(v, ai). i

Note that X is a deformation retract of its neighbourhood Nε(X), that ends the proof. Here we remark that at each stage of the proof of Theorem 5.10 one has X X χ(V α) = I(v, ai),

Vβ ⊂V α ai∈Vβ the ﬁrst summand being for all strata Vβ in V α, including Vα. In fact, our proof shows the more precise result:

Theorem 5.12 ([Sc4, Théorème6.2.2]) Let X be an analytic subset of the analytic manifold M and {V/α} a Whitney stratification of the pair (M,X). Let us denote by D a compact domain with a smooth boundary transverse to the strata. Let v (resp. v−) be a radial vector field pointing outwards (resp. inwards ∂D). There is a finite number of − zeroes ai of v (resp. v ) in D and we have : X X χ(X ∩ D) = I(v, ai) = I(v|V , ai) α(ai) ai∈X∩D ai∈X∩D X X − χ(X ∩ D) − χ(X ∩ ∂D) = χ(Vα ∩ int(D)) = I(v , ai)

Vα⊂X ai∈X∩D X − = I(v |V , ai), α(ai) ai∈X∩D

where, if dim Vα(a) = 0, then by convention I(v|Vα(a) , a) = +1.

46 6 Schwartz and MacPherson classes

In the following, M will be a complex analytic manifold equipped with an analytic strat- ¯ ˙ ¯ ification {Vi}: for every stratum Vi, the closure Vi and the boundary Vi = Vi \ Vi are analytic sets, union of strata. We denote by X ⊂ M a complex analytic compact subset stratified by {Vi}. The first definition of Chern class for singular varieties was given by M.H. Schwartz in the preprint [Sc1] (Lille University), then in 1965 in two “Notes aux CRAS” [Sc2]. Here we provide a sketch of the M.H. Schwartz construction.

6.1 Radial frames Let X ⊂ M be a singular n-dimensional complex variety embedded in a complex m- dimensional manifold. Let us consider a Whitney stratification {Vi} of M such that X is a union of strata and denote by (K) a triangulation of M compatible with the stratification, i.e. each open simplex is contained in a stratum. The first observation by M.-H. Schwartz concerns the triangulations: One knows (3.22) that the primary obstruction dimension to the construction of an r-frame tangent to M is 2p = 2(m − r + 1). In the same way, the primary obstruction dimension to the construction of an r-frame tangent to the 2s-dimensional stratum Vi is 2(s−r +1). That means that if one intends to construct a stratified r-frame tangent to X using the triangulation (K), then one will obtain obstruction cocycles on the strata with different dimensions according to the dimension of the considered stratum. Considering the triangulation (K), one cannot obtain a global cocycle with a well-defined dimension. The M.-H. Schwartz observation is the following: Let us denote by (D) the dual cell decomposition of (K) associated to a barycentric subdivision (K0) (see 1.13). Each (D)-cell is transverse to the strata. In particular, if d is a (D)-cell whose dimension 2p = 2(m − r + 1) is the obstruction dimension for the construction of an r-frame tangent to M and if Vi is a stratum of dimension 2s, then the dimension of the cell d ∩ Vi is

dim(d ∩ Vi) = 2(m − r + 1) − 2(m − s) = 2(s − r + 1) that is precisely the obstruction dimension for the construction of an r-frame tangent to Vi. This observation leads naturally to the construction of a stratified vector field by induction on the dimension of the strata, using the dual cell decomposition (D) and not the triangulation (K). The second observation by M.-H. Schwartz is that one has to consider stratified vector fields and frames which are radial in the sense we explained in the previous section. Below,

47 we provide an explicit construction of radial extension of frames.

The obstruction dimension for a r-frame, over M is equal to 2p = 2(m − r + 1). That means that one can construct such a section, without singularity over the (2p−1)-skeleton D(2p−1) of D and with isolated singularities over D(2p). 2s The obstruction dimension for an r-frame tangent to Vi is equal to 2q = 2(s − r + 1). That means that one considers strata such that s ≥ r − 1. As we know, the dual cell decomposition is transverse to the stratification, that means that if d2p is a 2p-dimensional 2s 2p 2s cell in D which intersects Vi , then d ∩Vi is a 2q-dimensional ∆ complex. Let us denote 2q 2p 2s ∆ = d ∩ Vi . (r) (r−1) r 2q 2s Let us consider a stratified r-frame v = (v , vr), section of E over ∆ ⊂ Vi , with isolated singularities which are zeroes of the last vector vr. One suppose that vr has 2q length less than 1. One can define in a tube Tε(∆ ) of radius ε (see above) on one hand (r) (r−1) (r) the parallel extensionv ˆ = (ˆv , vˆr) of v and on the other hand the radial transverse vector field τ and one has:

Proposition 6.1 (Radial extension for a frame) If µ and ε are sufficiently small, the (r) (r) (r−1) radial extension of v , defined by v˜ = (ˆv , vˆr + τ) satisfies the following conditions:

1. the radial extension of vr satisﬁes the Proposition 5.9, 2. if the (r − 1)-frame v(r−1) has no singularity on ∆2q and if v(r) admits an isolated 2q 2s (r) (r−1) singularity in a ∈ ∆ ∩ K ⊂ Vi which is a zero of vr, then v˜ = (ˆv , vˆr + τ) 2q satisﬁes the same properties in Tε(∆ ). In that case, if the (r−1)-complex plane generated by v(r−1)(a) is linearly independent 2q 2s (r) of the tangent plane T (∆ , a) in T (Vi , a), then the index of the extension v˜ in a, considered as an r-frame tangent to M is equal to the index of v(r) in a considered 2s as an r-frame tangent to Vi .

0 2s 3. In the same hypothesis than (ii), if q = 0 (i.e; s = r − 1), and if a = ∆ ⊂ Vi is a (r) zero of vr, then the index of v in a is +1.

We will denote by I(v(r), a) the index of v(r) in the isolated singularity a.

6.1.1 Global radial extension The “global” construction of vector fields by radial extension goes as follows: If the lowest dimensional stratum is a 0-dimensional one, i.e. a set of finitely many points ak, then one consider a radial vector field v in a ball B(ak) centered in each of these points. If the lowest dimensional of strata is a stratum Vi of dimension 2s > 0, then one construct a vector field v on Vi with isolated singularities. We notice that Vi is a manifold and it has to be compact if X is compact; in this case the total Poincaré-Hopfindex of v on Vi is χ(Vi). Now we go up by increasing dimensions of the strata containing Vi in their closure: if Vi ⊂ V j, then we extend v to a neighborhood of Vi in X as above, and then extend it further to all of Vj with isolated singularities. We proceed further in this way to get a stratified vector field v on all of X. Furthermore, by construction one has that if

48 we consider the closed strata V j, then the vector field is pointing inwards Vj near its boundary. Thus one has, by theorem , that the Poincaré-Hopfindex of v on each stratum Vj is χ(Vj). (r) 2q 2p 2s One will construct v over the subsets Ai = d ∩ Vi , by increasing dimensions of the strata Vi. One will construct at each step over Ai and a tube Tε(Ai), neighbourhood (2p−1) of Ai in D . At the following steps, the vector field could be modified, but without a tube Tε0 (Ai) ⊂ Tε(Ai). 2r−2 i) If Vi is a stratum whose real dimension is 2r − 2 = 2(m − p), the obstruction dimension to the construction of a section of Vr(TVi) is zero. One takes any (r − 1)-frame (r−1) 2r−2 0 2p 2r−2 v tangent to Vi in the vertices aj = ∆j of ∆ located in d ∩ Vi and vr zero in these points. 0 One construct the radial extension of the r-vector in the tubes Tε(∆j ) as a r-frame (r) (r) still denoted by v . According to Proposition 6.1 (iii), one has I(v , aj) = +1. ii) Let us suppose s > r − 1 and the construction already performed on all strata Vk whose dimension is less than 2s. That means that the construction has been performed on the sets Ak and the tubes Tε(Ak). Let us consider a 2s-dimensional stratum Vi. ˙ Let us denote Ai = Ai ∩ (Vi \ Vi). The r-frame is already constructed over Ai ∩ ˙ Tε(Ai). One can extend it on the rest of Ai with isolated singularities denoted by ak ans 2q ˙ located in 2q-open cells ∆k located outside T1(Ai) (see[Sc4]) and such that, over a ball 2q bk neighbourhood of ak in ∆k , one has: (r−1) 2r−2 2q a) v (x) generates a (r − 1)-complex plane P (x) supplementary of T (∆k , x) in 2s T (Vi , x), 2q b) for x ∈ Bk \{ak}, one has vr(x) tangent to ∆k and it has length less than 1. (r) ˙ At that stage, one has v on Ai ∪ Tε(Ai). One can extend it in a tube Tε0 (Ai) (with 0 ε < ε1 < ε), in such a way that one has: ˙ ˙ • the r-frame does not change within a tube Tε1 (Ai) ⊂ Tε(Ai), ˙ (r) • in Tε0 (Ai) ⊂ Tε(Ai), the extension is the radial extension of v , ˙ ˙ • in Tε(Ai) ⊂ Tε1 (Ai), the obtained r-frame is linear combination of the r-frame v(r)| already constructed and of the radial extension of v(r)| . Tε(A˙i) Ai

The constructed r-frame satisﬁes the properties of

Theorem 6.2 [B-S], [Sc2], [Sc5] One can construct, on the 2p-skeleton (D)2p, a stratiﬁed r-frame v(r), called radial frame, whose singularities satisfy the following properties: (r) (i) v has only isolated singular points, which are zeroes of the last vector vr. On (D)2p−1, the r-frame v(r) has no singular point and on (D)2p the (r − 1)-frame v(r−1) has no singular point. 2p (r) (ii) Let a ∈ Vi ∩ (D) be a singular point of v in the 2s-dimensional stratum Vi. If s > r − 1, the index of v(r) at a, denoted by I(v(r), a), is the same as the index of the (r) 2p restriction of v to Vi ∩ (D) considered as an r-frame tangent to Vi. If s = r − 1, then I(v(r), a) = +1. (iii) Inside a 2p-cell d which meets several strata, the only singularities of v(r) are inside the lowest dimensional one (in fact located in the barycenter of d).

49 (iv) The r-frame v(r) is pointing outwards a (particular) regular neighborhood U of X in M. It has no singularity on ∂U.

6.2 Schwartz classes Let us denote by T the tubular neighborhood of X in M consisting of the (D)-cells which meet X. Let us recall that d∗(σ) is the elementary (D)-cochain whose value is 1 at d(σ) and 0 at all other cells. We can deﬁne a 2p-dimensional (D)-cochain in C2p(T , ∂T ) by: X c˜ = I(v(r), σˆ) d∗(σ). d(σ)∈T dim d(σ)=2p

In other words, the cochainc ˜ satisﬁes

hc˜ · d(σ)i = I(v(r),, σˆ).

In a classical way the cochain is a cocycle, the obstruction cocycle (see [Sc4]) whose class cp(X) lies in

H2p(T , ∂T ) ∼= H2p(T , T\ X) ∼= H2p(M,M \ X), where the ﬁrst isomorphism is given by retraction along the rays of T and the second by excision (by M \T ).

Deﬁnition 6.3 [Sc2],[Sc5] The p-th Schwartz class is the class

cp(X) ∈ H2p(M,M \ X).

M.-H. Schwartz proved that the class does not depend of the diﬀerent choices involved in its construction. The proof of this fact is now easier, using Theorem 6.24 below.

6.3 Nash transformation Let M be an complex analytic manifold, of complex dimension m. Let X be a n- dimensional subanalytic complex variety, X ⊂ M. Let us denote by Σ = Xsing the singular part of X and by Xreg = X \ Σ its regular part. m m The Grassmanian manifold of complex n-planes in C is denoted by Gn(C ). Let us consider the Grassmann bundle of n (complex) planes in TM, denoted by G. The ﬁbre m Gx over x ∈ M is the set of n-planes in Tx(M), isomorphic to Gn(C ). An element of G is denoted by (x, P ) where x ∈ M and P ∈ Gx. On the regular part of X, one can deﬁne the Gauss map γ : Xreg −→ G by

γ(x) = (x, Tx(Xreg)).

Deﬁnition 6.4 The Nash transformation Xe is deﬁned as the closure of the image of γ in G.

50 G Xe = Imγ ,→ G % γ ↓ ν ↓ ↓ (6.5) Xreg ,→ MX,→ M

In general, Xe is not smooth, nevertheless, it is an analytic variety and the restriction ν : Xe → X of the bundle projection G → M is analytic. Let us denote by E the tautological bundle over G. The ﬁbre EP at a point (x, P ) ∈ G is the set of the vectors v in the n-plane P ∈ Gx.

EP = {v(x) ∈ TxM : v(x) ∈ P } Let us deﬁne E = E| , then E| can be identiﬁed with T (X ) where X = e X˜ e X˜reg reg ereg −1 ∼ ν (Xreg) = Xreg and

Ee = E ×G Xe = {(v(x), x˜) ∈ E × Xe : v(x) ∈ x˜} x˜ ∈ Xe is a n-complex plane in Tx(M) and x = ν(˜x). One has a diagram: Ee −−−→ E     y y Xe −−−→ G   ν  y y X −−−→ M An element in Ee is written (x, P, v) where x ∈ X, P is a n-plane in ν−1(x) and v is a vector in P . If x ∈ Xreg, then P = Tx(Xreg). Let us denote by {Vi} a complex analytic stratification of (M,X) satisfying the Whit- ney conditions. The following lemma is fundamental for the understanding of the geometrical definition of the local Euler obstruction. We recall the proof which is a direct application of the Whitney condition (a). Lemma 6.6 ([B-S, Proposition 9.1]) A stratified vector field v on A ⊂ X admits a canonical lifting v˜ on ν−1(A) as a section of Ee.

ν∗ Ee −→ TM|X v˜ ↑↓ v ↑↓ ν∗(x, x,˜ v(x)) = (x, v(x)). ν Xe −→ X

Proof: Let us consider a stratified vector field v on A ⊂ X and a point xe ∈ Xe. Let us denote x = ν(xe). −1 (i) If x ∈ Xreg, then v(x) ∈ Tx(Xreg) = xe with xe = ν (x). We definev ˜(˜x) = (x, Tx(Xreg), v(x)). −1 (ii) If x ∈ Vi, then v(x) ∈ Tx(Vi). Eachx ˜ ∈ ν (x) is in the closure of the image of γ, i.e. there is a sequence (xen) of points of Xereg such thatx ˜ = lim xfn, ν(xfn) = xn ∈ Xreg and x = T (W ). Then one has lim(x ) = x and lim T (X ) =x ˜. By the Whitney fn xn reg n xn reg condition (a), one has Tx(Vi) ⊂ x˜ and we can definev ˜(˜x) = (x, x,˜ v(x)).

51 6.4 Mather classes We introduce now two ingredients useful to deﬁne the MacPherson classes, namely the Mather classes and the local Euler obstruction. Mather classes are deﬁned as Chern classes of the Nash bundle, more precisely:

Deﬁnition 6.7 Let X ⊂ M a singular algebraic complex subvariety of a complex algebraic manifold M. The Mather class of X is deﬁned by:

∗ cM (X) = ν∗(c (Ee) ∩ [Xe]) where c∗(Ee) denotes the usual (total) Chern class of the bundle Ee in H∗(Xe) and the cap-product with [Xe] is the Poincar´eduality homomorphism. Let us recall that in general, Poincar´ehomomorphism is not an isomorphism. The Mather classes do not satisfy the Deligne-Grothendieck’s conjecture that we state later on (see Theorem 6.13).

6.5 Euler local obstruction The notion of local Euler obstruction was defined originally by R. MacPherson [MP] in 1974. Definitions equivalent to MacPherson’s have been given by several authors. It has been shown in [BDK] that the local invariant of singularities which appear in the Kashiwara formula for the index of holonomic modules [Ka] is equal to the local Euler obstruction. The original definition, due to R. MacPherson [MP] uses differential form. We will introduce the dual definition, using vector fields. Notation Let us consider a bundle E over X ⊂ Cm and A ⊂ X with boundary ∂A. Let us suppose that s is a section of E defined over the ∂A. The obstruction cocycle to extend the section s inside A will be denoted by Obs(s, E, A). m Let z = (z1, . . . , zm) be local coordinates in C around {0}, such that zi(0) = 0, we m denote by Bε and Sε the ball and the sphere centered in {0} with radius ε in C . Let us −1 −1 denote by Oν (Bε),ν (Sε) the orientation class (fundamental class) −1 −1 −1 −1 Oν (Bε),ν (Sε) ∈ H2n(ν (Bε), ν (Sε); Z). Let us recall that a radial vector field v in a neighborhood of the point {0} ∈ X is a stratified vector field so that there exists ε0 > 0 such that for all ε, 0 < ε < ε0, the vector v(x) is pointing outwards the ball Bε over the boundary Sε = ∂Bε. By the Bertini-Sard theorem, Sε is transverse to the strata Vi if ε is small enough, so the definition takes sense.

Definition 6.8 [B-S] Let v be a stratified radial vector field over X ∩ Sε andv ˜ the lifting −1 of v on ν (X ∩ Sε). The local Euler obstruction Eu0(X) is the obstruction to extend ˜ −1 v˜ as a nowhere zero section of E over ν (X ∩ Bε), evaluated on the orientation class −1 −1 Oν (Bε),ν (Sε): −1 Eu0(X) = Obs(˜v, E,e ν (X ∩ Bε)). The local Euler obstruction is independent of all choices involved.

52 6.5.1 Properties of Euler local obstruction Theorem 6.9 [MP],[GS] The Euler obstruction satisﬁes:

1. Eux(X) = 1 if x is a regular point of X;

0 0 2. Eux×x0 (X × X ) = Eux(X) · Eux0 (X );

3. If X is a curve, then Eux(X) is the multiplicity of X at x; If X is the cone over a non singular plane curve of degree d and x is the vertex of 2 the cone, then Eux(X) = 2d − d ;

4. If X is locally reducible at x and Xi are its irreducible components, then Eux(X) = P Eux(Xi).

The following proposition has been proved by many authors, in particular see [B-S] :

Proposition 6.10 (Constructibility) ([MP],[B-S],[Du] and other authors): The local Eu- ler obstruction is constant along the strata of a Whitney stratiﬁcation of X.

The main property of local Euler obstruction is the following Brasselet-Schwartz Pro- portionality Theorem [B-S]. Things being local, one can suppose we are in a ball Bε in m C local chart of M, with center a ∈ Vi ⊂ X. Theorem 6.11 ([B-S], Théorème 11.1) (Proportionality Theorem for vector fields). Let v be a stratified vector field obtained by radial extension (of a vector field on Vi) with an isolated singularity in the point a ∈ Vi, with index I(v, a) = I(v|Vi , a), then

−1 Obs(˜v, E,e ν (Bε ∩ X)) = Eua(X) · I(v, a).

One deﬁnes the bundle of frames in Ee in an obvious way. We will denote by Eer the bundle of r-frames in Ee.

Theorem 6.12 ([B-S], Th´eor`eme11.1) (Proportionality Theorem for frames). Let vr be a radial r-frame with an isolated singularities at the barycenter of the 2p-cells d2p = d(σ) with index I(vr, σˆ) at {σˆ} = d2p ∩ σ. Then the obstruction to the extension of v˜r as a section of Eer on ν−1(d2p ∩ X) is

r r −1 2p r Obs(˜v , Ee , ν (d ∩ X)) = Euσˆ(X) · I(v , σˆ).

6.6 MacPherson classes Let us recall firstly some basic definitions. A constructible set in a variety X is a subset obtained by finitely many unions, intersections and complements of subvarieties. A constructible function α : X → Z is a function such that α−1(n) is a constructible set for all n. The constructible functions on X form a group denoted by F(X). If A ⊂ X is a subvariety, we denote by 1A the characteristic function whose value is 1 over A and 0 elsewhere.

53 If X is triangulable, α is a constructible function if and only if there is a triangulation (K) of X such that α is constant on the interior of each simplex of (K). Such a triangulation of X is called α-adapted. The correspondence F : X → F(X) deﬁnes a contravariant functor when considering the usual pull-back f ∗ : F(Y ) → F(X) for a morphism f : X → Y . One interesting fact is that it can be made a covariant functor when considering the pushforward deﬁned on characteristic functions by: −1 f∗(1A)(y) = χ(f (y) ∩ A), y ∈ Y for a morphism f : X → Y , and linearly extended to elements of F(X). The following result was conjectured by Deligne and Grothendieck in 1969. Theorem 6.13 [MP] Let F be the covariant functor of constructible functions and let H∗(; Z) be the usual covariant Z-homology functor. Then there exists a unique natural transformation c∗ : F → H∗(; Z) ∗ satisfying c∗(1X ) = c (X) ∩ [X] if X is a manifold. The theorem means that for every algebraic complex variety, one has a natural transformation c∗ : F(X) → H∗(X; Z) satisfying the following properties:

1. c∗(α + β) = c∗(α) + c∗(β) for α and β in F(X),

2. c∗(f∗α) = f∗(c∗(α)) for f : X → Y morphism of algebraic varieties and α ∈ F(X), ∗ 3. c∗(1X ) = c (X) ∩ [X] if X is a manifold. The MacPherson’s construction uses both the constructions of Mather classes and local Euler obstruction. Proposition 6.14 There is a isomorphism T between algebraic cycles on X and constructible functions, given by X X T ( niVi)(p) = niEup(Vi) For a Whitney stratiﬁcation, we have the following lemma:

Lemma 6.15 [MP] There are integers nα such that, for every point x ∈ X, we have: X nαEux(Vα) = 1. α Proof: The proof is an easy exercise.

Deﬁnition 6.16 [MP] The MacPherson class of X is deﬁned by X c∗(X) = c∗(1X ) = nα i∗cM (Vα) α where i denotes the inclusion Vα ,→ X.

Note that we have the following relation : cM (X) = c∗(EuX ). In particular, the Chern classes of an algebraic variety are represented by algebraic cycles.

54 6.7 Schwartz and MacPherson classes In this section, we show the Theorem 6.24 that states that Schwartz and MacPherson classes correspond via Alexander duality (see section 5.2.1).

Proposition 6.17 The are a simplicial triangulation K of M compatible with the strat- iﬁcation and a cellular decomposition K˜ of the Grassman bundle G compatible with the −1 strata ν (Vi) such that: 1. equipped with the triangulation K (resp. with the decomposition K˜ ), M (resp. G) is a combinatorial variety,

2. the triangulation of M and decomposition of G are C1-diﬀerentiable,

3. for each cell σ˜β, then ν(˜σβ) is a simplex σα,

4. the restriction of ν to each cell σ˜β has constant rank.

˜ One says that the K-cellσ ˜β is horizontal if one has dim ν(˜σβ) = dimσ ˜β. In that case, the restriction of ν on the (open) cellσ ˜β is a diﬀeomorphism on ν(˜σβ). In the sequel, one deﬁnes a simplicial subdivision ∆˜ of K˜ . Image of a ∆-simplex˜ will not be a ∆-simplex but will satisfy a property that is given in the following Proposition (see [B-S]).

Proposition 6.18 . One can construct a simplicial subdivision ∆˜ of K˜ satisfying the following conditions: ˜ • (i) Every open cell σ˜β contains one and only one ∆-vertex denoted by a˜β. Let aα be −1 the barycenter of σα = ν(˜σβ), then a˜β is contained in the inverse image ν (aα), ˜ • (ii) Let σ˜β be a ∆-cell with image σα, and σαi a simplex in the boundary of σα. Let −1 ˜ us denote by d(σαi ) the dual cell of σαi , then σ˜β ∩ ν (σαi ) is a ∆-complex.

From the construction of ∆,˜ one has the following results:

m m Proposition 6.19 Let σα a m-dimensional K-simplex, d(σα ) the dual cell in M. Let m ˜ m h us denote by σ˜β the horizontal K-cells whose image is σα and d(˜σβ ) their dual cells in G. Then one has

−1 m [ h ˜ Closure of ν (d(σα ) ∩ X) = Closure of d(˜σβ ) ∩ X. (6.20) β

−1 Corollary 6.21 For each q-dimensional cell d(σα), one has ν (d(σα)) is a q-dimensional ∆˜ -complex.

2r−2 2r−2 Let σα be a (2r − 2)-dimensional K-simplex, d(σα ) its dual cell whose inter- −1 section with X is 2p-dimensional. It results from Corollary 6.21 that ν (d(σα)) is a 2p-dimensional ∆-complex.˜ One can proves the following Theorem:

55 Theorem 6.22 Let c˜ be a ∆˜ -cocycle of the 2p-Chern class cp(E˜) of the Nash bundle E˜. −1 2p Let us denote kα = hc,˜ ν (Dα )i. Then the Mather class cM,r−1(X) contains the cycle

X 2r−2 kασα 2r−2 σα ⊂X

˜ ˜2n Proof: Let us consider the ∆-cyclew ˜2n which is the sum of all simplexes δ with the ˜ ˜ orientation induced by the one of X. The class ofw ˜2n in H2n(X) is the fundamental class [X˜]. Letc ˜ be a cocycle in C2p(X˜) representing the Chern class cp(E˜), then a cycle of the ∆˜ p ˜ ˜ Mather class cM,r−1(X) = ν∗(c (E) ∩ [X]) is given by

ν∗(˜c ∩ w˜2n). (6.23) Let us recall the result in [B1]: The Poincar´emorphism C2p(X˜) → C (X˜), cap- ∆˜ 2r−2,K˜ product byw ˜2n is composed of the Alexander isomorphism ([B1], §3) and the Thom ˜ ˜2t ˜ 2r−2 homomorphism, dual of intersection with X. In another words, if dβ = d(˜σβ ) is the 2r−2 dual cell, in G of the (2r − 2)-cellσ ˜β , then one has

X 2r−2 ˜2t ˜ c˜ ∩ w˜2n = νβσ˜β where νβ = hc˜ · dβ ∩ Xi. 2r−2 ˜ σ˜β ⊂X (see [B1], formulae (7), (8) and diagram (16)). 2r−2 Taking the image by ν∗ of that cycle, the contributions of horizontal cellsσ ˜β are the only one which do not vanish, the others having an image whose dimension is less than 2r − 2. The cycle 6.23 is homologous to the cycle

X 2r−2 X 2r−2 ν∗( νβσ˜β ) = kασα 2r−2 β σα ⊂X

P P ˜2t ˜ 2r−2 where kα = νβ = hc˜· dβ ∩ Xi, the sum being extended on indices β such thatσ ˜β 2r−2 is horizontal with image σα . One obtains the result, using Proposition 6.19. The following result has been proved in [B-S]:

Theorem 6.24 [B-S] The MacPherson class is the image of the Schwartz class by the Alexander duality isomorphism

2(m−r+1) =∼ H (M,M \ X) −→ H2(r−1)(X).

Proof: Using the notations of section 6.2, the r-frame vr determines a cocycle of the M.H. Schwartz class:

X 2q ∗ X r bc = να(dα ) with να = I(v , aj). (6.25) 2q 2q dα ∩X6=∅ aj ∈dα ∩X

It determines also a cocyclec ˜ of the Chern class cp(E˜) such that

−1 2q < c.ν˜ (dα ∩ X) >= Euaα (X)µα,

56 2r−2 2q 2q 2r−2 where aα is any point of σα , simplex whose the cell dα is dual, i.e. dα = d(σα ). M Theorem 6.22 implies that the Chern-Mather class cr−1(X) is represented by the cocycle: X 2r−2 Euaα (X)µασα 2r−2 σα ⊂X 2q 2r−2 where µα is the coeﬃcient of the cocyclec ˆ relatively to the cell dα dual of σα and 2r−2 where aα is any point of σα . M Let us write the previous result for each V i: The Chern-Mather class cr−1(V i) is represented by the cocycle:

X 2r−2 Euaα (V i)µασα . 2r−2 σα ⊂V i

By deﬁnition 6.16, the MacPherson class c∗(X) is represented by the cocycle:

X X 2r−2 ni Euaα (V i)µασα . i 2r−2 σα ⊂V i

2r−2 In this expression, the coeﬃcient of µασα is

X 2r−2 Cα = niEuaα (V i) = 1, with Iα = {i : σα ⊂ V i} = {i : aα ∈ V i}. i∈Iα One obtains a cycle of the MacPherson class of X of the form:

X 2r−2 γ = µασα . 2r−2 σα ⊂X

2q Let us recall (5.2.1) that the Alexander isomorphism H (M,M − X) → H2r−2(X) is induced by the isomorphism: 2q ˚ C(D)(M,M − T ) → C2r−2,(K)(X)

2q ∗ 2q 2r−2 which associates to a (D)-cochain (dα ) such that dα ∩ X 6= ∅ the (K)-chain σα such 2q 2r−2 that dα = d(σα ). By this isomorphism, the cycle γ is image of the cocycle of the M.H.Schwartz class (cf 6.25), which proves the theorem. We observe that we determined a cocycle of the MacPherson class. In fact, one has the following corollary: Corollary 6.26 Let (K) be a simplicial triangulation of M compatible with a Whitney stratiﬁcation and vr a r-radial frame deﬁned on, the 2q-squeleton D(2q) of a cellular decomposition (D) dual of (K). The (r − 1)-st MacPherson class cr−1(X) is represented by the cycle X I(v(r), σˆ) σ σ∈X where dim σ = 2(r − 1).

57 6.8 Schwartz-MacPherson classes for projective cones In this section, we show the following result, due to Barthel, Brasselet and Fieseler [BBF] following ideas of Brasselet and Gonzalez-Sprinberg [BG].

Theorem 6.27 Let Y ⊂ PN , be a projective variety and ı: Y,→ KY the canonical inclusion in the projective cone KY on Y with vertex {s}. Let us denote also by K : H∗(Y ) → H∗+2(KY ) the homological projective cone, one has

cj(KY ) = ı∗cj(Y ) + Kcj−1(Y ), (6.28) where Kc−1(Y ) denotes the class [s] ∈ H0(KY ).

Let us consider an n-dimensional projective variety Y in PN and let us denote by L the restriction of the hyperplane bundle of PN to Y . If H denotes an hyperplane in PN , then D = Y ∩ H is a divisor in Y and we denote 2 by [D] ∈ H2n−2(Y ) the fundamental class. By Poincaréduality, let ηD ∈ H (Y ) be the 1 associated cohomology class. L is the bundle associated to D and one has c (L) = ηD. We denote by E the completed projective space of the total space of L, i.e. P(L ⊕ 1Y ) where 1Y is the trivial bundle of complex rank 1 on Y . The canonical projection p: E → Y admits two sections, zero and infinite, with images Y(0) and Y(∞). The projective cone KY is obtained as a quotient of E by contraction of Y(∞) in a point {s}. It is the Thom space associated to the bundle L, with basis Y . Let us consider p: E → Y as a sphere bundle with fibre S2, subbundle of a bundle ¯ 3 3 ¯ p¯: E → Y with fibre the ball B . We denote by θE¯ ∈ H (E,E) the associated Thom class; One has a Gysin exact sequence

γ pj ... → Hj+1(Y ) → Hj−2(Y ) → Hj(E) → Hj(Y ) → ... ; in which the gysin map γ is the composition of

−1 −1 (¯pj−2) ¯ (∩θE¯ ) ¯ ∂ Hj−2(Y ) −→ Hj−2(E) −→ Hj+1(E,E) → Hj(E) and can be explicited in the following way: If ζ is a cycle representing the class [ζ] of −1 Hj−2(Y ), then γ([ζ]) is the class of the cycle p (ζ) in Hj(E). Let π the canonical projection π : E → KY .

Deﬁnition 6.29 We call homological projective cone and we denote by κ the composition κ = π∗γ : Hj−2(Y ) → Hj(KY ) for j ≥ 2. We let κ(0) := [s] ∈ H0(KY ) for 0 = H−2(Y ).

Let us remark that for j 6= 0, κ is an homomorphism. The theorem 6.27 is a direct consequence of the following proposition proved in [BBF].

Proposition 6.30 The Chern classes of E and Y are related by the formula

c∗(E) = (1 + η0 + η∞) ∩ γ(c∗(Y )), (6.31)

1 2 where ηj := c O(Y(j)) ∈ H (E) for j = 0, ∞, and ∩ denotes the usual cap-product.

58 Proof of the Theorem 6.27 (from the Proposition 6.30). Let 1E the constructible function which is the characteristic function of E, then one has

( χ(Y ), if x = s π∗(1E)(x) = 1, elsewhere, i.e.

π∗(1E) = 1KY + (χ(Y ) − 1)1{s}. As one has

π∗c∗(1E) = c∗(π∗(1E))

one obtains

π∗c∗(E) = c∗(KY ) + (χ(Y ) − 1) [s]. (6.32)

On another hand, from the formula (6.31) one obtains:

π∗c∗(E) = π∗γ(c∗−1(Y )) + π∗(η0 ∩ γ(c∗(Y ))) + π∗(η∞ ∩ γ(c∗(Y ))). (6.33)

Let ι0 : Y,→ E and ι∞ : Y,→ E be the inclusions of Y as zero and inﬁnite sections of E respectively. By deﬁnition of γ, for a cycle ζ in Y and for j = 0 or ∞, one has

ηj ∩ γ([ζ]) = (ιj)∗([ζ])

then

π∗(ηj ∩ γ(c∗(Y ))) = π∗ιj∗c∗(1Y ) = π∗c∗(1Y(j) ) = c∗π∗(1Y(j) ).

Let us denote by ι = π ◦ ι0 : Y,→ KY the natural inclusion of Y in KY , one has

π∗(1Y(0) ) = 1ι(Y ) and π∗(1Y(∞) ) = χ(Y )1{s}.

One obtains

π∗(η0 ∩ γ(c∗(Y ))) = c∗(1ι(Y )) = ι∗c∗(Y ), and

π∗(η∞ ∩ γ(c∗(Y ))) = χ(Y )c∗(1{s}) = χ(Y )[s],

where [s] is the class of the vertex s in H0(KY ). The comparison of the formulae (6.32) and (6.33) gives:

c∗(KY ) = ı∗c∗(Y ) + π∗γc∗−1(Y ) + [s],

and the Theorem 6.27.

59 6.9 Schwartz-MacPherson classes of Thom spaces associated to embeddings The previous construction associates canonically a Thom space to the embedding of a smooth variety Y in PN . As examples, let us consider the image of the Segre embedding P1 × P1 ,→ P3, deﬁned in homogeneous coordinates by

ϕS :(x0 : x1) × (y0 : y1) 7→ (x0y0 : x0y1 : x1y0 : x1y1),

1 1 That is an embedding whose bidegree is (1, 1). Image ϕS(P × P ) is a non degenerate quadric Q provided with two families of generatices. Euler class of the bundle E in 2 1 1 2 1 2 1 1 H (Px × Py) = H (Px) ⊕ H (Py) = Z ⊕ Z is c (E) = (ηx, ηy) where ηx is Euler class of 1 1 the hyperplane bundle of Px, i.e. such that ηx ∩ [Px] = 1. The image of the Veronese embedding P2 ,→ P5 deﬁned by

2 2 2 ϕV :(x0 : x1 : x2) 7→ (x0 : x0x1 : x0x2 : x1 : x1x2 : x2).

2 That is an embedding whose degree is 2. Image ϕV (P ) is smooth and has degree 4. It is 2 2 1 called Veronese surface. Euler class of the bundle E in H (P ) is c (E) = 2ηK where H 5 2 ∼ being hyperplane in P , one has H ∩ V is a divisor in P = 2K, K being hyperplane in 2 1 2 2 P . Then ηK = c (EK ) is generator of H (P ), one has ηK ∩ [K] = 1. With the previous construction, KY is the Thom space associated to the ﬁbre bundle L, of complex rank 1 and restriction to Y of the hyperplane bundle of PN . Chern classes and intersection homology of these exemples have been computed in [BG]. In the case of the Segre embedding, let d1 and d2 two ﬁxed lines belonging each to a system of generatrices of the quadric Y = P1 × P1. Let us denote by ω the canonical generator of 2 1 ∗ 1 1 1 H (P ), one has c (P ) = 1 + 2ω and c∗(P ) = [P ] + 2??. Then

1 1 c∗(Y ) = c∗(P × P ) = ([Y ] + 2[d1]) ∗ ([Y ] + 2[d2]) = [Y ] + 2([d1] + [d2]) + 4[a] where a is a point in Y and where ∗ denotes the intersection of cycles or homology classes. One has κ(c∗(Y )) = [KY ] + 2([Kd1] + [Kd2]) + 4[Ka]. let us denote by ∼ the homology relation of cycles. In KY , one has [BG], §3:

Y ∼ Kd1 + Kd2, d1 ∼ d2 ∼ Ka, a ∼ s, and, with 6.27

c∗(KY ) = [KY ] + 3([Kd1] + [Kd2]) + 8[Ka] + 5[s] , | {z } | {z } | {z } |{z} H6(KY ) H4(KY ) H2(KY ) H0(KY ) which is the result of [BG]. In the case of the Veronese embedding, let d be a projective line in Y := P2, one has: c∗(P2) = 1 + 3ω + 3ω2 where ω is the canonical generator of H2(P2), and is dual, by 2 Poincar´eisomorphism of the class [d] ∈ H2(P ). One has, by Poincar´eduality

c∗(Y ) = [Y ] + 3[d] + 3[a]

60 where a is a point in Y . One has

K(c∗(Y )) = [KY ] + 3[Kd] + 3[Ka] such that, in KY , [BG], §3.b, Y ∼ 2Kd, d ∼ 2Ka and a ∼ s. One has

c∗(KY ) = [KY ] + 5[Kd] + 9[Ka] + 4[s]. | {z } | {z } | {z } |{z} H6(KY ) H4(KY ) H2(KY ) H0(KY )

61 7 Other classes and comparisons

7.1 Fulton classes The Fulton classes ([Fu] exemple 4.2.6 (a)) and Fulton-Johnson classes ([FJ], [Fu] exemple 4.2.6 (c)) have been deﬁned in a general framework. In the case of Local Complete Intersection, these classes coincide and can be deﬁned in the following (simpler) way:

If X is a local complete intersection, then the normal bundle of Xreg in M extends canonically to X as a vector bundle NX M and the Fulton class is equal to

F −1 c (X) = c(TM|X )c(NX M) ∩ [X] = c(τX ) ∩ [X].

Here τX = TM|X − NX M denotes the virtual tangent bundle on X, deﬁned in the Grothendieck group of vector bundles on X. If X is a manifold, then cF (X) is the usual Chern class c∗(X). Let M be a non-singular compact complex analytic variety of pure dimension n + 1 and let L be a holomorphic line bundle on M. Take f ∈ H0(M,L), a holomorphic section of L, such that the variety X of zeroes of f is a (nowhere dense) hypersurface in M. Then, the Fulton class of X is F c (X) = c(TM|X − L|X ) ∩ [X]. In [BLSS1], the authors show the following result:

Theorem 7.1 Let us assume that X ⊂ M is a hypersurface, defined by X = f −1(0), where f : M → D is a holomorphic function into an open disc around 0 in C. For each point a ∈ X, let Fa denote a local Milnor fiber, and let χ(Fa) be its Euler-Poincaré FJ characteristic. Then the Fulton-Johnson class cr−1(X) of X of degree (r−1) is represented in H2(r−1)(X) by the cycle

X r χ(Fσˆα ) I(v , σˆα) · σα (7.2)

σα⊂X, dim σα=2(r−1)

7.2 Milnor classes The comparison between the Schwartz-MacPherson classes and the Fulton-Johnson classes can be viewed in two ways, which coincide in some classical situations. In the case of isolated singularities, the diﬀerence between Schwartz-MacPherson classes and Fulton-Johnson classes is given by the Milnor numbers at the singular points:

62 Theorem 7.3 [SS] If X is compact and the singularities of X are isolated points {xi} where X is a local complete intersection. Then

q F n+1 X c∗(X) − c (X) = (−1) µxi [xi] ∈ H0(X). i=1

This motivates the following deﬁnition given by various authors:

Deﬁnition 7.4 ([A1],[BLSS1],[PP2], [Yo]) The diﬀerence class

n F µ∗(X) = (−1) (c (X) − c∗(X)) is called the Milnor class of X.

7.2.1 Description in terms of constructible functions The following description comes from [PP2]. Let us come back to the hypersurface situation: M is a non-singular compact complex analytic variety of pure dimension n + 1 and L is a holomorphic line bundle on M. The hypersurface X in M is the set of zeroes of a holomorphic section of L. Consider the function χ : X → Z defined by χ(x) := χ(Fx), where Fx denotes the Milnor fibre at x and χ(Fx) its Euler characteristic. Define also the function µ : X → Z n−1 by µ = (−1) (χ − 1X ). Fix any stratification S of X such that µ is constant on the strata of S, for instance a Whitney stratification of X. The topological type of the Milnor fibre is constant along the strata of the Whitney stratification of Z. Let us denote by µS the value of µ on the stratum S. Let X 0 α(S) = µS − α(S ) S06=S,S⊂S0 be the numbers defined inductively on descending dimensions of S.

Theorem 7.5 [PP2] We have

X −1 −1 µ∗(X) = α(S)c(L|X ) ∩ (iS,X )∗c∗(S) = c(L|X ) ∩ c∗(µ), S∈S where iS,X : S → X denotes the natural inclusion.

The formula was conjectured in [Yo] when X is projective. Under this last assumption, [PP1] proved earlier that Z Z X −1 µ∗(X) = α(S) c(L|S) ∩ c∗(S) X S∈S S

63 7.3 Motivic Chern classes: Hirzebruch theory for singular varieties In the same way that the MacPherson Chern functor generalises the Chern class, the Todd class and the Thom-Hirzebruch class have been generalised as natural transformations respectively by Baum-Fulton-MacPherson and by Cappell-Shaneson. We show that the motivic theory allows to unify the three generalisations in the case of singular varieties. In the case of singular varieties, the three classes (Chern Todd and L-class) have been generalized as natural transformations of functors, in the following way: Deﬁnition 7.6 [Chern Transformation (MacPherson)] (see [MP]) Theorem 6.13 tells us that there is an unique natural transformation

c∗ : F(X) → H∗(X) from the group of constructible functions F(X) to homology, satisfying the suitable properties. In particular for the constructible function 1X , one deﬁnes c∗(X) := c∗(1X ): Schwartz-MacPherson class of X. Deﬁnition 7.7 [Todd Transformation (Baum-Fulton-MacPherson)] [BFM] There is an unique natural transformation

td∗ : G0(X) → H∗(X) ⊗ Q from the Grothendieck group of coherent sheaves on X, satisfying suitable axioms, in particular, for the structure sheaf OX on a smooth variety, td∗(OX ) is the Todd class of X. In general, one deﬁnes td∗(X) := td∗([OX ]) as being the Baum-Fulton-MacPherson Todd class of X. Deﬁnition 7.8 [L-Transformation (Cappell-Shaneson)] [CS1, CS2] There is an unique natural transformation

L∗ : Ω(X) → H2∗(X; Q) from the group of constructible self-dual sheaves on X, satisfying suitable axioms, in particular, for the intersection sheaf ICX on a smooth variety, L∗([ICX ]) is the L-class of X. In general, one deﬁnes L∗(X) := L∗([ICX ]) as being the Cappell-Shaneson L-class of X. In short, one has the following table:

X manifold X singular variety number cohomology classes homology classes χ(X) Chern Schwartz-MacPherson — — —

χa(X) Todd Baum-Fulton-MacPherson — — — sign(X) Thom-Hirzebruch Cappell-Shaneson

64 The problem is that the three transformations are deﬁned on diﬀerent spaces:

F(X),G0(X) and Ω(X) and one asks for the possibility of unifying in the same way thatn the Hirzebruch theory in the smooth case. The problem has been solved (see [BSY]) using the motivic framework. Let us recall some ingredients which will be useful:

Deﬁnition 7.9 Let X be an algebraic variety, the Grothendieck relative group of algebraic varieties over X denoted by K0(var/X) is the quotient of the free abelian group of isomorphy classes of algebraic maps Y −→ X, modulo the “additivity relation”:

[Y −→ X] = [Z −→ Y −→ X] + [Y \ Z −→ Y −→ X] for closed algebraic sub-spaces Z in Y .

In [BSY], the authors prove the following 4 theorems:

Theorem 7.10 The map

e : K0(var/X) −→ F(X) deﬁned by e([f : Y → X]) := f!1Y is the unique group morphism which commutes with direct images for proper maps and such that e([idX ]) = 1X for X smooth and pure dimensional.

Theorem 7.11 There is an unique group morphism

mC : K0(var/X) −→ G0(X) which commutes with direct images for proper maps and such that

mC([idX ]) = [OX ] for X smooth and pure dimensional.

Theorem 7.12 The morphism sd : K0(var/X) −→ Ω(X) deﬁned by sd([f : Y → X]) := [Rf∗QY [dimC(Y )+dimC(X)]] is the unique group morphism which commutes with direct images for proper maps and such that

sd([idX ]) = [QX [2 dimC(X)]] = [ICX ] for X smooth and pure dimensional.

65 Theorem 7.13 There is an unique group morphism

Ty : K0(var/X) −→ H∗(X) ⊗ Q[y] which commutes with direct images for proper maps and such that

Ty([idX ]) = tdg(y)(TX) ∩ [X] for X smooth and pure dimensional.

In particular, one has: T−1([idX ]) = c∗(X).

Remark 7.14 If a complex algebraic variety X has only rational singularities (for example if X is a toric variety), then:

mC([idX ]) = [OX ] ∈ G0(X) and in this case T0([idX ]) = td∗(X).

That is not true in general !

The main result is the following:

Theorem 7.15 One has a commutative “tripode” diagramme:

e mC F(X) ←− K0(var/X) −→ G0(X) sd & c∗ ↓ Ty ↓ Ω(X) ↓ td∗

y=−1 y=0 H∗(X) ⊗ Q ←− H∗(X) ⊗ Q[y] L∗ ↓ −→ H∗(X) ⊗ Q. y=1 & H∗(X) ⊗ Q

7.4 Verdier Riemann-Roch Formula Theorem 7.16 Let f : X0 → X be a smooth map (or a map with constant relative dimension), then one has

∗ ∗ tdg(y)(Tf ) ∩ f Ty([Z −→ X]) = Tyf ([Z −→ X]).

0 Here Tf is the bundle over X of tangent spaces to ﬁbres of f.

Proposition 7.17 (Factorisation of Ty) Let us deﬁne P<∞ −i td(1+y)([F]) := i=0 tdfi([F]) · (1 + y) . Then one has:

Ty = td(1+y) ◦ mC : K0(var/X) −→ H∗(X) ⊗ Q[y].

66 References

[Al] J. Alexander, A proof of the invariance of certain constants in Analysis situs, Trans. Am. Math. Soc., 16 (1915), 148-154.

[A1] P. Aluﬃ, Chern classes for singular hypersurfaces, Trans. Amer. Math. Soc. 351 (1999) 3989–4026.

[BBF] G. Barthel, J.-P. Brasselet et K.-H. Fieseler, Classes de Chern des variétéstoriques singulières, C.R.A.S. t. 315, SérieI, p. 187-192, 1992.

[BFM] P. Baum, W. Fulton, R. MacPherson, Riemann-Roch for singular varieties. Publ. Math. I.H.E.S. 45, 101-145 (1975)

[Be] Betti Sopra gli spazi di un numero qualcunque di dimensioni, Ann. Math. Pura Appl. 2 (4), (1871) 140-158.

[B1] J.P. Brasselet Définitioncombinatoire des homomorphismes de Poincaré,Alexander et Thom pour une pseudo-variété, Astérisque82-83 (1981), 71-91.

[B2] J.P. Brasselet. Local Euler obstruction, old and new, XI Brazilian Topology Meeting, 1998, World Sci Publishing, 2000, 140–147.

[B3] J.-P. Brasselet, Characteristic Classes and Singular Varieties. Book to appear.

[BG] J.P. Brasselet et G. Gonzalez-Springberg. Espaces de Thom et contre-exemples de J.L. Verdier et M. Goresky, Bol. Soc. Brazil. Mat. 17 (1986), no 2, 23–50.

[BLSS1] J.-P. Brasselet, D. Lehmann, J. Seade, T. Suwa Milnor numbers and classes of local complete intersections, Proc. Japan Acad., 75, Ser. A (1999) 179-183.

[BLSS2] J.-P. Brasselet, D. Lehmann, J. Seade, T. Suwa Milnor classes of local complete intersections, Trans. Amer. Math. Soc. 354 (2002), no 4, 1351–1371.

[B-S] J.P. Brasselet, M.H. Schwartz: Sur les classes de Chern d’un ensemble analytique complexe, Ast´erisque82-83 (1981), 93-147.

[BSS] J.-P. Brasselet, J. Seade, T. Suwa, Livro, to appear.

[BSY] J.-P. Brasselet, J. Sch¨urmann, S. Yokura, Hirzebruch classes and motivic Chern classes for singular spaces, math.AG/0503492.

[BDK] J.L. Brylinski, A. Dubson, M. Kashiwara, Formule de l’indice pour les Modules Holonomes et obstruction d’Euler, C.R. Acad. Sc. Paris 293 s´erieA 1981, 573-576.

[CS1] S.E. Cappell and J.L.Shaneson, Stratiﬁable maps and topological invariants. J. Amer. Math. Soc. 4, 521-551 (1991)

[CS2] S.E. Cappell and J.L.Shaneson, Genera of algebraic varieties and counting lattice points. Bull. Amer. Math. Soc. 30, 62-69 (1994)

67 [Ch] S.S. Chern, Characteristic classes of hermitian manifold, Ann. Math. 47 (1946), 85-121.

[Di] J. Dieudonné, A History of Algebraic and Differential Topology, Birkhäuser(1988).

[Du] A. Dubson. Classes caractéristiquesdes variétéssingulières, C.R. Acad.Sc. Paris 287sérieA 1978, 237-240.

[Eu1] L. Euler Leonard Euler und Christian Goldbach, Briefwechsel, 1729-1764, herans- gegeben und eingeleitet von A.P. Juˇskeviˇcund E. Winter, Berlin Akademie-Verlag 1965.

[Eu2] L. Euler Demonstration nonularum insignium proprietatum, quibus solida hedris planis inclusa sunt praedita 1751. Novi Comment. acad. sc. Petrop. t.4 (1752-53) Saint P´etersbourg, 1758, p. 140-160.

[Fu] W. Fulton, Intersection Theory, Springer-Verlag, (1984).

[FJ] W. Fulton and K. Johnson, Canonical classes of singular varieties. Manuscripta Math. 32, no. 3-4 (1980), 381–389.

[GS] G. Gonzalez-Sprinberg. L’obstruction d’Euler locale et le th´eor`emede MacPherson, Asterisque 82-831981, 7-32.

[GP] V. Guillemin and A. Pollack Diﬀerential topology Prentice Hall (1974).

[HT] S. Halperin and Toledo, Stiefel Whitney Homology Classes, Annals of Math 96 (1972),

[He] P. Heegaard, Forstudier til en topologisk Teori for de algebraiske Fladers Sam- menhng, Dissertation, Copenhagen 1898.

[HeF] P. Heegaard, Sur l’analysis situs, Bull. Soc. Math. France 44 (1916) 161 - 242. Translation.

[Hirs] M.W. Hirsch Diﬀerential Topology Graduate Texts in Mathematics, 33, New York, Springer-Verlag, 1976.

[Hirz] F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer-Verlag, 1966.

[Ho] H. Hopf, Vektorfelden in n-dimensionalen Mannigfaltikeiten, Math. Annalen 96 (1927), 225-250.

[Hu] D. Husemoller, Fibre bundles, Graduate texts in Mathematics no 20, Springer- Verlag, 1966.

[Ka] M. Kashiwara, Index theorem for a maximally overdetermined system of linear differential equations, Proc. Japan Acad. 49 (1973), 803-804.

[Le] S. Lefschetz, Algebraic Topology

68 [Li] E. Lima A caracter´ıstica de Euler-Poincaré MatemáticaUniversitária,Sociedade Mathematica Brasileira, No 1, Junho 1985, pp 47–62.

[Lo] S. Lojasiewicz, Triangulation of semi-analytic sets, Annali Sc. Norm. Sup. de Pisa 18 (1964), 449-474.

[MP] R.MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. 100, no 2 (1974), 423-432.

[Mi1] J. Milnor, Topology from the Diﬀerentiable Viewpoint , Univ. Press of Virginia, Charlottesville, 1965.

[Mi2] J. Milnor, Singular points of complex hypersurfaces, Ann. of Math. Studies 61, Princeton 1968.

[MS] J. Milnor and J. Stasheﬀ, Characteristic Classes, Princeton University Press (1974).

[PP1] A. Parusi´nskiand P. Pragacz, A formula for the Euler characteristic of singular hypersurfaces, J. Alg. Geom., 4 (1995), 337-351.

[PP2] A. Parusi´nskiand P. Pragacz, Characteristic classes of hypersurfaces and characteristic cycles, J. Alg. Geom., 10 (2001), no 1, 63–79.

[Po1] H. Poincaré Mémoire sur les courbes définiespar une équationdifférentielle, Jour. Math. Pures et Appl. (4)1, (1885), 167-244.

[Po2] H. Poincaré Sur la généralisation d’un théorèmed’Euler relatif aux polyèdres, CRAS vol. 117 (1893) 144.

[Po3] H. Poincar´e Analysis situs J. Ecole Polytechnique, t. 1, 1895, p. 1-121

[Po4] H. Poincaré Complémentàl’Analysis situs Rend. Circ. Matem. Palermo, t. 13, 1899, p. 285-343

[Po5] H. Poincaré Second complémentàl’Analysis situs Proc. London Math. Soc., t. 32, 1900, p. 277-308

[Sc1] M.H.Schwartz, Classes obstructrices d’un sous-ensemble analytique complexe d’une variétélisse. Lille 1964, second version in Publ. de l’U.F.R. de Mathématiques de Lille, 11, 1986.

[Sc2] M.H.Schwartz, Classes caractéristiquesdéfiniespar une stratification d’une variété analytique complexe, CRAS 260, (1965), 3262-3264 et 3535-3537.

[Sc3] M.H. Schwartz: Champs radiaux et préradiaux associésàune stratification, CRAS 303 (1986), no 6, 239–241.

[Sc4] M.H. Schwartz: Champs radiaux sur une stratiﬁcation analytique, Travaux en cours, 39 (1991), Hermann, Paris.

[Sc5] M.-H. Schwartz, Classes obstructrices des ensembles analytiques. 2001.

69 [SS] J. Seade and T. Suwa, An adjunction formula for local complete intersections Int. J. Math. 9, (1998), 759-768.

[Ste] N. Steenrod, The Topology of Fibre Bundles, Princeton Univ. Press (1951).

[Sti] E. Stiefel Richtungsfelder und fernparallelismus in Mannigfaltigkeiten Comm. Math. Helv., 8, (1936) 3-51.

[Su1] T. Suwa, Classes de Chern des intersections compl`eteslocales, C.R.Acad.Sci. Paris, 324, (1996), 67-70.

[Su2] T. Suwa, Dual class of a subvariety, Tokyo J. Math. 23 (2000), no 1, 51–68.

[Tr] D. Trotman Book in preparation on stratiﬁcations.

[Va] I. Vainsencher Classes caracter´ısticas em Geometria Alg´ebrica IMPA, Rio, 1985.

[Wh1] H. Whitney Sphere spaces Proc. Nat. Acad. Sci., 21, 462-468, (1935).

[Wh2] H. Whitney On the Theory of Sphere Bundles, Proc. Nat. Acad. Sci. USA 26 (1940).

[Yo] S. Yokura, On a Milnor class, Preprint (1997).

[Zh] J. Zhou, Classes de Wu et classes de Mather, C. R. Acad. Sci. Paris, S´er.I, 319, (1994), no. 2, 171-174.