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 $f_\ast([M]) \neq 0 \in H_n(K(\pi_1(M,\ast),1),\mathbb{Z})$.

Continue reading «Essential manifolds»

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.

Continue reading «Aspherical manifolds»

Now this is a slightly corrected (although still somewhat messy) version of my diploma thesis - in german:
Matsumotos Satz und A¹-Homotopietheorie.

You can read something about the content in this blog post, containing an extended abstract in english.

Continue reading «Diploma thesis (in german)»

Nearly half of all blog posts start with "Soon this blog will be full of content" and the other half with "it has been incredibly silent, soon I will post a lot". This post is of the second type, but I apologize by linking to what has kept me from writing here:
Extended abstract of my diploma thesis.
While the diploma thesis is in german, the abstract is in english and only 9 pages long, without any proofs. The diploma thesis is now available here (updated on 2011-08-09).

As a teaser, here is the abstract of the abstract:

In classical covering space theory we have an isomorphism of the fundamental group with the fibre of the universal cover over the basepoint. Covering spaces of topological groups are group extensions, but not every group extension is a covering space. Perfect groups admit a universal central extension and the kernel of this extension is also called fundamental group. For simply connected Chevalley-groups over a perfect field, this fundamental group, classically called second unstable K-Theory, is exactly the fundamental group of a simplicial resolution. The loops are described explicitly by matrices.

We look at the model structure Voevodsky and Morel use in their 1999 IHES paper and discuss 1.2, 1.3, 1.4, 1.5, 1.6, 1.8, 1.9, 1.10. There is nothing difficult or particularly interesting, but you might want to look up some specific issue or reference.
Continue reading «Walk-through to Morel-Voevodsky A¹-homotopy theory, page 48-50»

I decided to post some background needed in order to understand Morel-Voevodsky's paper "A¹-homotopy theory". I explain some notions of simplicial sets, topoi, monoidal categories, enriched categories and simplicial model categories.

I tried to give many more references I found useful.
Continue reading «Categorical background for A¹-homotopy theory (simplicial model categories)»