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.

Continue reading «All these fundamental groups!»

Last week Matthias Wendt and I have put our first joint paper on the arXiv, it's called "A¹-fundamental groups of isotropic reductive groups" and here I will tell you how you can think about it without going through all definitions.

I have blogged about this before, but until a few days ago, there was only my diploma thesis in german and an extended abstract in english online, and the results were not as general (oh, and the proofs were also not the best...).

Continue reading «A¹-fundamental groups of isotropic groups»

Every simplicial set admits a universal covering space. If the simplicial set is a simplicial group, the covering can be given a group structure, such that the covering map is a group homomorphism. This can be done "at once" without using geometric realizations, just by close analysis of the usual construction of the universal covering.

Let $G_\bullet$ be a simplicial group, $PG_\bullet$ its path space, $\Omega G_\bullet$ its loop space and $N_\bullet$ the kernel of the projection $\Omega G_\bullet \rightarrow \pi_1(G_\bullet, id)$, which consists of all contractible loops. The composition of $N_\bullet \hookrightarrow \Omega G_\bullet$ with the inclusion $\Omega G_\bullet \hookrightarrow PG_\bullet$ yields a monomorphism $N_\bullet \hookrightarrow PG_\bullet$ whose cokernel we denote by $\tilde{G}_\bullet$.

I claim that this simplicial group already does the job.
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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»