Algebraic models for rational G

... Schwede and Shipley in [SS03b] provided a quite general tool for establishing a Quillen equivalence between a spectral model category with a set of (homotopically) compact, cofibrant and fibrant generators and a certain category of modules over a ring with possibly many objects. This machinery can b ...

... Schwede and Shipley in [SS03b] provided a quite general tool for establishing a Quillen equivalence between a spectral model category with a set of (homotopically) compact, cofibrant and fibrant generators and a certain category of modules over a ring with possibly many objects. This machinery can b ...

Abelian Sheaves

... In Chapter 4 we study abelian sheaves on topological spaces. We introduce the functors ( • )Z and ΓZ ( • ) associated to a locally closed subset Z and we study flabby sheaves. Then we study locally constant abelian sheaves. We prove that the cohomology of such sheaves is a homotopy invariant, and us ...

... In Chapter 4 we study abelian sheaves on topological spaces. We introduce the functors ( • )Z and ΓZ ( • ) associated to a locally closed subset Z and we study flabby sheaves. Then we study locally constant abelian sheaves. We prove that the cohomology of such sheaves is a homotopy invariant, and us ...

pdf

... be an introduction to K-theory, both algebraic and topological, with emphasis on their interconnections. While a wide range of topics is covered, an effort has been made to keep the exposition as elementary and self-contained as possible. Since its beginning in the celebrated work of Grothendieck on ...

... be an introduction to K-theory, both algebraic and topological, with emphasis on their interconnections. While a wide range of topics is covered, an effort has been made to keep the exposition as elementary and self-contained as possible. Since its beginning in the celebrated work of Grothendieck on ...

barmakthesis.pdf

... That means that for any two points of X0 there exists an open set which contains only one of them. Therefore, when studying homotopy types of finite spaces, we can restrict our attention to T0 -spaces. In [37], Stong defines the notion of linear and colinear points, which we call up beat and down be ...

... That means that for any two points of X0 there exists an open set which contains only one of them. Therefore, when studying homotopy types of finite spaces, we can restrict our attention to T0 -spaces. In [37], Stong defines the notion of linear and colinear points, which we call up beat and down be ...

Elementary Number Theory - science.uu.nl project csg

... Most of us have heard about them at a very early age. We also learnt that there are inﬁnitely many of them and that every integer can be written in a unique way as a product of primes. These are properties that are not mentioned in our rules. So one has to prove them, which turns out to be not entir ...

... Most of us have heard about them at a very early age. We also learnt that there are inﬁnitely many of them and that every integer can be written in a unique way as a product of primes. These are properties that are not mentioned in our rules. So one has to prove them, which turns out to be not entir ...

Kazhdan`s Property (T)

... a natural question, since Property (T) implies finite generation (Kazhdan’s observation) but not finite presentation (as shown by examples discovered later). There is also a section on Kostant’s result according to which the isometry group Sp(n, 1) of a quaternionic hyperbolic space (n ≥ 2) has Prop ...

... a natural question, since Property (T) implies finite generation (Kazhdan’s observation) but not finite presentation (as shown by examples discovered later). There is also a section on Kostant’s result according to which the isometry group Sp(n, 1) of a quaternionic hyperbolic space (n ≥ 2) has Prop ...

Introduction to Combinatorial Homotopy Theory

... Homotopy theory is a subdomain of topology where, instead of considering the category of topological spaces and continuous maps, you prefer to consider as morphisms only the continuous maps up to homotopy, a notion precisely defined in these notes in Section 4. Roughly speaking, you decide not to di ...

... Homotopy theory is a subdomain of topology where, instead of considering the category of topological spaces and continuous maps, you prefer to consider as morphisms only the continuous maps up to homotopy, a notion precisely defined in these notes in Section 4. Roughly speaking, you decide not to di ...

# Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f mapping a compact convex set into itself there is a point x0 such that f(x0) = x0. The simplest forms of Brouwer's theorem are for continuous functions f from a closed interval I in the real numbers to itself or from a closed disk D to itself. A more general form than the latter is for continuous functions from a convex compact subset K of Euclidean space to itself.Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics.In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem and the Borsuk–Ulam theorem.This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry.It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.The theorem was first studied in view of work on differential equations by the French mathematicians around Poincaré and Picard.Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods.This work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by Jacques Hadamard and by Luitzen Egbertus Jan Brouwer.