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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign 'quantities' to the chains of homology theory.From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century; from the initial idea of homology as a topologically invariant relation on chains, the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra. The terminology tends to mask the fact that in many applications cohomology, a contravariant theory, is more natural than homology. At a basic level this has to do with functions and pullbacks in geometric situations: given spaces X and Y, and some kind of function F on Y, for any mapping f : X → Y composition with f gives rise to a function F o f on X. Cohomology groups often also have a natural product, the cup product, which gives them a ring structure. Because of this feature, cohomology is a stronger invariant than homology, as it can differentiate between certain algebraic objects that homology cannot.