All these fundamental groups!

Tuesday, April 02nd, 2013 | Author:

There are a lot of fundamental groups floating around in mathematics. This is an attempt to collect some of the most popular and sketch their relations to each other.

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Motives of Projective Bundles

Friday, December 14th, 2012 | Author:

Given a vector bundle E-->X of rank r+1 one can take the projective space of lines in each fiber, which results in a projective bundle P(E)-->X. A projective bundle formula for a functor F from spaces to rings tells us that F(P(E)) is a free F(X)-module of rank r.

In this post I look at some computations around projective bundle formulae for the Chow ring, the algebraic K-Theory and the (Chow) motive of some spaces, in particular flag varieties. We recover some results from the previous posts on cohomology, cycles & bundles and motive of projective space.

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Model categories

Monday, November 12th, 2012 | Author:

This is supposed to be a short intuitive introduction to model categories.

Suppose you have a category \mathcal{C} and some class of morphisms W which behave somewhat like isomorphisms (for example: Chain complexes and Quasi-isomorphisms, or topological spaces and homotopy equivalences, or simplicial sets and weak homotopy equivalences ...). We will call this class "weak equivalences". Then you can look at the localized category [W^{-1}]\mathcal{C}, where the morphisms in W are made invertible. If you're lucky, not all objects are isomorphic to each other, and if you're really lucky, you can compute something.

But, as it turns out, usually you don't work with the localized category abstractly, but by some explicit construction of some special case (say, Verdier localization of triangulated categories in the homological setting or explicit homotopies in the topological setting).

Model categories (and its cousins, weak factorization systems, categories of (co)fibrant objects, homotopical categories, etc.) provide a framework to compute stuff in [W^{-1}]\mathcal{C}.

[UPDATE 2013-03-06] I gave a 30-Minute talk about model categories, with very little content. [/UPDATE]

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Category: English, Mathematics | 2 Comments

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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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.

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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Gute Mathematik Skripten und Bücher

Wednesday, November 11th, 2009 | Author:

Ich mach mal eine Liste von den Grundlagen, und woher ich sie gelernt habe (=was ich als brauchbare Lektüre empfinde, also meine Lieblingsliteratur)
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Category: German, Mathematics | 3 Comments