Problem Set 5 - Stony Brook Mathematics
... Problem 1. Show that if X is a finite simplicial complex whose underlying topological
space is a homology n-manifold, then
(a) X consists entirely of n-simplices and their faces,
(b) Every (n − 1)-simplex is a face of precisely two n-simplices.
Problem 2. Suppose that X is a compact triangulable hom ...
Complex Bordism (Lecture 5)
... the cohomology theory MU of complex bordism. In fact, we will show that MU is universal among complexoriented cohomology theories.
We begin with a general discussion of orientations. Let X be a topological space and let ζ be a vector
bundle of rank n on X. We may assume without loss of generality th ...
LECTURE NOTES 4: CECH COHOMOLOGY 1
... If G is any old abelian group, then we can consider it as a discrete space. In that case, the sheaf U 7→
G(U ) = T (U, G) is called the constant sheaf with values in G.
The very constant presheaf with values in G is the presheaf G! whose value on an open set U is
G!(U ) = G.
(Check that this is not ...
... (invariants) of topological spaces
specifically we will be dealing with cubical sets
• “…Allows one to draw conclusions about
global properties of spaces and maps from
local computations.” (Mischaikow)
... where the product ranges over all (n + 1)-tuples i0 , . . . , in for which Ui0 ...in is non-empty. This is a
larger complex with much redundant information (the group A(Ui0 ...in ) now occurs (n + 1)! times),
hence less convenient for calculations, but it computes the same cohomology as we will now ...
Algebraic Topology Introduction
... Frequently quotient spaces arise as equivalence relations, as any surjective map of sets p : X → Y defines
an equivalence relation on X by x1 ∼ x2 if and only if p(x1 ) = p(x2 ). In general, if R is an equivalence
relation on X, then the quotient space X/R is the set of all equivalence classes of X ...
Sheaf Cohomology 1. Computing by acyclic resolutions
... sheaf, Čech cohomology (with respect to that cover) and right-derived functor cohomology are the same.
Note that the first proposition here holds without any assumption that the sheaf in question be flasque.
The acyclicity of the cover gives us a long exact sequence of the Čech groups, allowing an ...
AAG, LECTURE 13 If 0 → F 1 → F2 → F3 → 0 is a short exact
... (1) in the category of vector spaces over a field k all objects are injective;
(2) Q and Q/Z are injective in the category Ab of abelian groups;
(3) if R is a commutative ring then HomAb (R, Q/Z) is an injective R-module.
Note that the abelian groups Q and Q/Z are not finitely generated, and the
MANIFOLDS, COHOMOLOGY, AND SHEAVES
... To understand these lectures, it is essential to know some point-set topology,
as in [3, Appendix A], and to have a passing acquaintance with the exterior
calculus of differential forms on a Euclidean space, as in [3, Sections 1–4]. To be
consistent with Eduardo Cattani’s lectures at this summer sch ...
... Brouwer’s Fixed Point Theorem. A continuous map f : Bd+1 → Bd+1
has at least one fixed point x = f (x).
Proof. Let A, B : Sd → Sd be maps defined by A(x) = (x − f (x))/kx − f (x)k
and B(x) = x. B is the identity and therefore has degree 1. If f has no fixed
point then A is well defined and has degre ...
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.