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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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.

Continue reading «The four functors of Grothendieck in examples»

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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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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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.

Continue reading «A manifold whose functions are the smooth functions on the real line with rational period»

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:

• Definitions
• Examples and counter-examples
• Topological classification, the genus
• Digression about algebraic field extensions
• Biholomorphic classification via universal covering deck transformations
• Brief discussion of Gaussian curvature and conformal geometry
• Summary

Continue reading «Classifying Riemann surfaces»