Two postings: A new question and a followup to yesterday's comments.........DMD ____________________________________________ Subject: A question on cohomology and homology of some homogeneous spaces From: Dmitry Kerner Date: Fri, 25 Feb 2005 05:08:05 -0800 (PST) Dear topologists, I have a question on the cohomology ring and representatives of homology classes of the following spaces. Consider the space of (complex, nonzero) $k\times n$ matrices (projectivised, i.e. taken up to multiplication by scalar matrix). The group O(k) (or U(k)) acts on this space (by multiplication from the left). Factor by the action of this group. Where can I find (any information) about cohomology ring of this space, the explicit representatives of homology classes etc. The same question, if instead of O(k) (U(k)) we factor by a group of k by k (nonzero) matrices with left lower block (of dimension $r \times k-r$) consisting of zeros only. Any references or hints on explicit solution are welcome. ___________________________________________________ Subject: van Kampen From: "Ronald Brown" Date: Thu, 24 Feb 2005 22:08:02 -0000 reply to r.brown@bangor.ac.uk The papers mentioned by Gustavo do not discuss the fundamental groupoid on a set of base points. Of course for a simplicial set, or CW-complex, there is a distinguished set of base points, namely the vertices, but this disguises what one wants to do in calculations, namely to take a set of base points appropriate to the geometry of the union. It is also useful to have a proof of the 1-dimensional theorem which generalises to dimension 2 and above, so allowing calculation of for example some 2-types, or higher. For a recent survey on part of this, see `Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems', Proceedings of the Fields Institute Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories, September 23-28, Fields Institute Communications 43 (2004) 101-130. It may be useful to mention also the use of the fundamental groupoid for group actions: for a discontinuous action of a group G on a Hausdorff space X which admits a universal cover, the fundamental groupoid of the orbit space is the orbit groupoid of the fundamental groupoid. Again, this can be used as a mode of calculation, for example of the fundamental group of the symmetric square of a space. See preprint 02.25 on http://www.bangor.ac.uk/~mas010/brownpr.html as a revised version of a book chapter. Ronnie Brown www.bangor.ac.uk/~mas010 >> >> Subject: comment on Van Kampen >> From: Gustavo Granja >> Date: Thu, 24 Feb 2005 13:25:51 +0000 (WET) >> >> It seems to me that the best way of thinking about the Van Kampen >> theorem for arbitrary covers is the following: >> >> 1. Taking the fundamental groupoid is a left adjoint in a Quillen pair >> and hence commutes with homotopy colimits. >> >> 2. The homotopy colimit of the Cech diagram of an open cover is the space >> inquestion. (For a very readable recent treatment generalizing Segal's >> result see D. Dugger and D. Isaksen, "Topological hypercovers and >> A1-realizations" >> Math. Zeit. 246 #4 (2004) 667--689) >> >> 3. There is a very simple formula (amounting to the usual descent category >> in the case when the diagram is the nerve of a cover) for the homotopy >> colimit of a diagram of groupoids in >> >> Sharon Hollander "A homotopy theory for stacks", available on Hopf. >> >> See also >> >> Dror Farjoun, Emmanuel "Fundamental group of homotopy colimits." >> Adv. Math. 182 (2004), no. 1, 1--27. (Reviewer: Donald M. Davis) >> >> for a slightly different perspective. >> >> Gustavo Granja >> _________________________________________________________________