Thom spaces

Wednesday, June 13th, 2012 | Author:

I want to discuss the elementary basics of Thom spaces of vector bundles. To start, I explain general one-point compactification and a different construction on vector spaces, then I do it for vector bundles to define the Thom space. I also discuss suspension of topological spaces and how adding a trivial vector space (or bundle) corresponds to suspension under the forming of Thom spaces.

Motto: (Thom space:Suspension)::(Vector bundle:Trivial bundle) or

"Thom spaces are twisted suspensions"

Honestly, I really want to talk about algebraic (motivic) Thom spaces some day, but these are some preliminaries to understand what's going on, so I want to get this out first.

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The four functors of Grothendieck in examples

Tuesday, May 01st, 2012 | Author:

This post will discuss the definition of the four functors "pushforward" f_\ast, "pullback" f^\ast, "pushforward with compact support" f_{!} and "exceptional pullback" f^{!} of sheaves of abelian groups, associated to a continuous morphism f : X \to Y of topological spaces X and Y. Then we will look at maps f which are open immersions or closed immersions, and calculate in the example of \mathbb{C}^\times \to \mathbb{C} and its closed complement \{0\} \to \mathbb{C} exactly what happens. This is intended to give some intuition what the general four functor calculus is about.

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Essential manifolds

Saturday, August 13th, 2011 | Author:

Now I'll explain a little bit what essential manifolds are and what they're good for.

Definition
A (connected closed orientable topological) n-manifold M is called essential, if there exists a continuous map f : M \to K(\pi_1(M,\ast),1) such that the induced morphism on the top homology f_\ast : H_n(M,\mathbb{Z}) \to H_n(K(\pi_1(M,\ast),1),\mathbb{Z}) maps the fundamental class [M] \in H_n(M,\mathbb{Z}) to some non-zero element f_\ast([M]) \neq 0 \in H_n(K(\pi_1(M,\ast),1),\mathbb{Z}).

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Aspherical manifolds

Wednesday, August 10th, 2011 | Author:

In this post I want to sketch the idea of aspherical manifolds - manifolds which don't admit higher homotopically non-trivial spheres - and the related concepts of Eilenberg-MacLane-spaces and classifying spaces for groups.

Definition
A topological space M is called aspherical if all higher homotopy groups vanish, i.e. \pi_n(M,m_0) = 0 \quad \forall n > 1 where m_0 \in M is an arbitrary basepoint and M is assumed to be connected.

Since manifolds admit universal covers, you could equivalently define a manifold to be aspherical if and only if its universal cover is contractible.

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A manifold whose functions are the smooth functions on the real line with rational period

Wednesday, March 31st, 2010 | Author:

Hi, I was reading in

Jet Nestruev: Smooth Manifolds and Observables, Springer, 2003

about a month ago (after I stumbled over a question on MO) and there was an exercise that resisted solution for more than a week.

Well.... now I found out that I have just misread the exercise. However, this way I basically did several exercises at once. Here comes the problem and its solution:

The problem

(inspired by page 28, chapter 3, exercise 3.17.5 in Nestruev)

Find a smooth (real) manifold M such that its algebra of smooth functions C^\infty(M,\mathbb R) is isomorphic to the algebra of all smooth functions f : \mathbb R \to \mathbb R that have some rational period \tau (i.e. there exists \tau \in \mathbb Q such that f(x)=f(x+\tau) for all x). Note that we don't fix a period \tau here. Let's call the algebra in question (smooth functions on the real line with some rational period) A.

You might want to stop reading here and think for a second (or minutes) about the solution or similar problems that have easier solutions. A more vague problem would be

Find a space M such that the functions M \to \mathbb R correspond to functions \mathbb R \to \mathbb R that are periodic with some rational period.

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Classifying Riemann surfaces

Wednesday, October 21st, 2009 | Author:

In this post, I will sketch a classification of Riemann surfaces.

For those who haven't heard about the subject before, there is an introduction. For the impatient, look at the bottom of the post, where I have written a very short summary.

Table of contents:

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