By Jean-Paul Brasselet

Vector fields on manifolds play an immense function in arithmetic and different sciences. specifically, the Poincaré-Hopf index theorem supplies upward thrust to the speculation of Chern sessions, key manifold-invariants in geometry and topology.

It is traditional to invite what's the ‘good’ inspiration of the index of a vector box, and of Chern sessions, if the underlying area turns into singular. The query has been explored by means of numerous authors leading to a number of solutions, beginning with the pioneering paintings of M.-H. Schwartz and R. MacPherson.

We current those notions within the framework of the obstruction conception and the Chern-Weil idea. The interaction among those equipment is without doubt one of the major positive aspects of the monograph.

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**Vector fields on singular varieties**

Vector fields on manifolds play an enormous position in arithmetic and different sciences. particularly, the Poincaré-Hopf index theorem offers upward thrust to the speculation of Chern sessions, key manifold-invariants in geometry and topology. it's average to invite what's the ‘good’ idea of the index of a vector box, and of Chern periods, if the underlying area turns into singular.

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3) For each pair (Vα , Vβ ) such that Vα ∩ V β = ∅ one has Vα ⊂ V β . The highest dimensional stratum, which may not be connected, is called the regular stratum and usually denoted by V0 or Vreg . 2. A stratiﬁcation (Vα )α∈A of V is said to be Whitney if it further satisﬁes the following two conditions, known as the Whitney conditions (a) and (b), for every pair (Vα , Vβ ) such that Vα ⊂ V β . Let xi ∈ Vβ be an arbitrary sequence converging to some point y ∈ Vα and yi ∈ Vα a sequence that also converges to y ∈ Vα .

3) cq (T , ∂T ; v (r) ) ∈ H 2q (T , ∂T ). 4. The Poincar´e–Hopf class of v (r) at S, which is denoted by PH(v(r) , S), is the image of cq (T , ∂T ; v (r) ) by the isomorphism H 2q (T , ∂T ) H 2q (T , T \ S) followed by the Alexander duality (see [25]) ∼ AM : H 2q (T , T \ S) −→ H2r−2 (S). 3). Note that if dim S < 2r − 2, then PH(v(r) , S) = 0. The relation between the Poincar´e–Hopf class of v(r) and the index we deﬁned above is the following: PH(v (r) , S) = Ind(v(r) , d(σ)) σ , where the sum runs over the 2(r − 1)-simplices σ of the triangulation of S and d(σ) is the dual cell of σ (of dimension 2q).

Let M be an m -dimensional oriented manifold (not necessarily compact) and S a compact subset of M . Let U0 = M \ S and let U1 be an open neighborhood of S. We consider the covering U = {U0 , U1 } of M . We set Ap (U, U0 ) = { ξ = (ξ0 , ξ1 , ξ01 ) ∈ Ap (U) | ξ0 = 0 }. Then we see that if ξ is in Ap (U, U0 ), Dξ is in Ap+1 (U, U0 ). This gives rise ˇ to another complex, called the relative Cech-de Rham complex, and we may ˇ deﬁne the p-th relative Cech-de Rham cohomology of the pair (U, U0 ) as p HD (U, U0 ) = KerDp /ImDp−1 .