PDF version - University of Warwick

... It is not hard to see, from graphical considerations, that V –shaped permutations are precisely the same as allowable permutations. Thus the singular permutation homology groups for allowable permutations are precisely the intersection homology groups. Further it can be seen that, given an X̄ –allow ...

... It is not hard to see, from graphical considerations, that V –shaped permutations are precisely the same as allowable permutations. Thus the singular permutation homology groups for allowable permutations are precisely the intersection homology groups. Further it can be seen that, given an X̄ –allow ...

Section 3.2 - Cohomology of Sheaves

... the canonical natural equivalence HA,I (X, −) = HA,I 0 (X, −). This means that the A-module structure induced on H i (X, F ) is independent of the choice of resolutions on Mod(X). Definition 5. Let (X, OX ) be a ringed space and A = Γ(X, OX ). Fix an assignment of injective resolutions J to the obje ...

... the canonical natural equivalence HA,I (X, −) = HA,I 0 (X, −). This means that the A-module structure induced on H i (X, F ) is independent of the choice of resolutions on Mod(X). Definition 5. Let (X, OX ) be a ringed space and A = Γ(X, OX ). Fix an assignment of injective resolutions J to the obje ...

Homology Theory - Section de mathématiques

... since Hn (1, 1) = 0). By naturality of ∂ the lower squares in the diagram commute and so, since p∗ ◦ ∂ = 0 ◦ p∗ we have im ∂ ⊂ ker p∗ which proves the first part of the proposition (this actually proves more generally that the induced morphisms in the upper row are well-defined). For the second part ...

... since Hn (1, 1) = 0). By naturality of ∂ the lower squares in the diagram commute and so, since p∗ ◦ ∂ = 0 ◦ p∗ we have im ∂ ⊂ ker p∗ which proves the first part of the proposition (this actually proves more generally that the induced morphisms in the upper row are well-defined). For the second part ...

7. A1 -homotopy theory 7.1. Closed model categories. We begin with

... which is the identity on objects and whose set of morphisms from X to Y equals the set of homotopy classes of morphisms from some fibrant/cofibrant replacement of X to some fibrant/cofibrant replacement of Y : HomHo(C) (X, Y ) = π(RQX, RQY ). If F : C → D if a functor with the property that F sends ...

... which is the identity on objects and whose set of morphisms from X to Y equals the set of homotopy classes of morphisms from some fibrant/cofibrant replacement of X to some fibrant/cofibrant replacement of Y : HomHo(C) (X, Y ) = π(RQX, RQY ). If F : C → D if a functor with the property that F sends ...

introduction to algebraic topology and algebraic geometry

... These notes assemble the contents of the introductory courses I have been giving at SISSA since 1995/96. Originally the course was intended as introduction to (complex) algebraic geometry for students with an education in theoretical physics, to help them to master the basic algebraic geometric tool ...

... These notes assemble the contents of the introductory courses I have been giving at SISSA since 1995/96. Originally the course was intended as introduction to (complex) algebraic geometry for students with an education in theoretical physics, to help them to master the basic algebraic geometric tool ...

1 An introduction to homotopy theory

... defines a homotopy of paths γ0 ⇒ γ1 . Hence there is a single homotopy class of paths joining p, q, and so Π1 (X) maps homeomorphically via the source and target maps (s, t) to X × X, and the groupoid law is (x, y) ◦ (y, z) = (x, z). This is called the pair groupoid over X. The fundamental group π1 ...

... defines a homotopy of paths γ0 ⇒ γ1 . Hence there is a single homotopy class of paths joining p, q, and so Π1 (X) maps homeomorphically via the source and target maps (s, t) to X × X, and the groupoid law is (x, y) ◦ (y, z) = (x, z). This is called the pair groupoid over X. The fundamental group π1 ...