By M. Boucetta, J.M. Morvan

The purpose of this quantity is to provide an advent and assessment to differential topology, differential geometry and computational geometry with an emphasis on a few interconnections among those 3 domain names of arithmetic. The chapters provide the heritage required to start examine in those fields or at their interfaces. They introduce new learn domain names and either outdated and new conjectures in those varied matters exhibit a few interplay among different sciences on the subject of arithmetic. issues mentioned are; the foundation of differential topology and combinatorial topology, the hyperlink among differential geometry and topology, Riemanian geometry (Levi-Civita connextion, curvature tensor, geodesic, completeness and curvature tensor), attribute sessions (to affiliate each fibre package deal with isomorphic fiber bundles), the hyperlink among differential geometry and the geometry of non soft gadgets, computational geometry and urban functions comparable to structural geology and graphism.

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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.