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Stefaan Vaes - Classification of regular subalgebras of the hyperfinite II1 factor

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Custom Filters release announcement. Related Question feed. There is also a more general version of the theorem for open covers consisting of an arbitrary number of open sets, but 1 the basic two-set version covers most of the applications, 2 the proof of the general version is pretty much the same as the version for two sets, only more difficult notationally. There is also a version for the fundamental groupoid on a subset of the basepoints.

It is actually this version that we are really after, since the whole fundamental groupoid is impractical in that it contains a lot of redundant information.

Our strategy for proving this version will be to deduce it from the version for the full fundamental groupoid. It only takes, as one says, a straight-forward diagram chase to prove that a retract of a pushout square must also be a pushout square. Now, in our case it is easy to show the second square is a retract of the first.

The retractions are built as follows. That is not an issue for the groupoid version. This makes no essential difference and only complicates the language, so we will silently use closed sets whenever we want, with the understanding that it should be checked that fattening them slightly will produce open sets that van Kampen applies to.

This simple idea has several applications:. We can also use the observation in the first paragraph to compute fundamental groups of connected sums of manifolds.

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Computing the fundamental groups of compact surfaces is easily done by starting with a construction of the surface as the result of identifying some sides of a polygon. Now we are going to prove the Jordan Curve Theorem.

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In Chapter 9 of Topology and Groupoids this construction is related to making identifications on a discrete subset of a topological space. These ideas usefully generalise to higher dimensions, via Higher Homotopy Seifert-van Kampen Theorems: see for example Part I of Nonabelian Algebraic Topology for results on second relative homotopy groups. There is more to be said Later : I realise I did not answer the question as to the purpose of the generalisation. As suggested, the immediate purpose was to have a theorem which yielded the fundamental group of the circle, which is, after all, THE basic example in algebraic topology.

PDF | On Jan 1, , W.B. Vasantha Kandasamy and others published Subset Groupoids. In mathematics, especially in category theory and homotopy theory, a groupoid generalises the .. If G = GL*(K), then the set of natural numbers is a proper subset of G0, since for each natural number n, there is a corresponding identity matrix.

Grothendieck in Section 2 of his "Esquisse d'un Programme" emphasises that choosing a single base point will often destroy any symmetry in the situation. Consider the following union of five open sets:. One is in a "Goldilock's" situation.

Choosing all points as base points is too large for comfort. Choosing one base point is too small. But choosing eight base points is just about right! Situations like this arise in combinatorial group theory.

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In general, one chooses the set of base points according to the geometry of the given situation. The proof given in the paper referred to is by verification of the universal property, and does not require knowledge that the category of groupoids admits colimits, nor any specific method of constructing them. The proof also has the advantage of generalising to higher dimensions, once one has the appropriate higher dimensional homotopical gadgets.

July 28, For further discussion on this area, see this mathoverflow discussion. Home Questions Tags Users Unanswered. Does May's version of groupoid Seifert-van Kampen need path connectivity as a hypothesis?

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