Topological Invariants of Stratified Spaces by Murray N. Rothbard

By Murray N. Rothbard

The imperative subject of this ebook is the recovery of Poincaré duality on stratified singular areas by utilizing Verdier-self-dual sheaves resembling the prototypical intersection chain sheaf on a posh variety.

After rigorously introducing sheaf conception, derived different types, Verdier duality, stratification theories, intersection homology, t-structures and perverse sheaves, the final word target is to give an explanation for the development in addition to algebraic and geometric houses of invariants corresponding to the signature and attribute periods effectuated by means of self-dual sheaves.

Highlights by no means earlier than awarded in publication shape contain entire and extremely certain proofs of decomposition theorems for self-dual sheaves, clarification of equipment for computing twisted attribute sessions and an advent to the author's thought of non-Witt areas and Lagrangian structures.

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If f ∈ HomC(A) (X • , Y • ) is null-homotopic with homotopy {s i }, then {F (s i )} is a null-homotopy for F (f ). Thus F induces a functor F : K(A) → K(B) between the homotopy categories. As an example, let X and Y be topological spaces and f : X → Y a continuous map, and take A = Sh(X), B = Sh(Y ). Then direct image f∗ : K(X) → K(Y ) and inverse image f ∗ : K(Y ) → K(X) are well-defined functors on the homotopy category of complexes of sheaves. 4 Derived Categories Let A be an abelian category.

To do this, we need to specify a translation automorphism T and a collection of distinguished triangles. Let T = [1], the shift on complexes. 3 The Triangulation of the Homotopy Category C n (u) = X n+1 ⊕ Y n , n 0 dX[1] n = dC(u) n+1 u dYn 33 . We have the natural inclusion i : Y • → C • (u) and projection p : C • (u) → The triangle X • [1]. p X • −→ Y • −→ C • (u) −→ X • [1] u i is called a standard triangle. Let Δ be the collection of all triangles in K(A) isomorphic to standard triangles. 1 The homotopy category K(A) together with the shift functor [1] : K(A) → K(A) and Δ as the collection of distinguished triangles forms a triangulated category.

We claim there is no quasiisomorphism t : Z • → X • with f t null-homotopic. Suppose we had such a t = {t n }n∈Z . Pick a cycle z ∈ Z 1 such that the class of t 1 (z) is the generator of H 1 (X • ) = Z/2. This implies that t 1 (z) is an odd integer. Now t 1 (2z) = 2t 1 (z) is even, so 0 in H 1 (X • ). As t is in Qis, it follows that the class of 2z is 0 in H 1 (Z • ) and 2z = dZ (z ) for some z ∈ Z 0 . 4 Derived Categories Z −1 dZ Z0 t0 s0 0 Z1 t1 s1 Z Z 1 2 1 Z 0 49 Z/3 We have 2t 1 (z) = t 1 (dZ (z )) = 2t 0 (z ) = 2s 1 (dZ (z )) and hence t 1 (z) = s 1 (dZ (z )) = s 1 (2z) = 2s 1 (z), contradicting t 1 (z) odd.

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