The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T07:37:13Z http://mathoverflow.net/feeds/question/39818 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39818/the-higher-van-kampen-theorems-and-computation-of-the-unstable-homotopy-groups-of The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres Harry Gindi 2010-09-24T04:05:06Z 2010-09-24T22:12:03Z <p>Since Ronnie Brown and his collaborators have come up with a general proof of the higher Van Kampen theorems, what impediments are there to using these to compute the unstable homotopy groups of spheres? </p> http://mathoverflow.net/questions/39818/the-higher-van-kampen-theorems-and-computation-of-the-unstable-homotopy-groups-of/39821#39821 Answer by David Roberts for The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres David Roberts 2010-09-24T04:45:26Z 2010-09-24T04:45:26Z <p>Ronnie and collaborators' HHvK theorems are essentially all for crossed complexes and similar. These are, as I'm sure you're aware, partially linearised homotopy types. In particular, for a simply connected space, they are just chain complexes. So for $S^k, k>1$ it's not more useful than the homology. </p> http://mathoverflow.net/questions/39818/the-higher-van-kampen-theorems-and-computation-of-the-unstable-homotopy-groups-of/39868#39868 Answer by Ronnie Brown for The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres Ronnie Brown 2010-09-24T15:21:28Z 2010-09-24T22:12:03Z <p>Here are some answers on the HHSvKT - I have been persuaded by a referee that we ought also to honour Seifert. </p> <p>These theorems are about homotopy invariants of structured spaces, more particularly filtered spaces or n-cubes of spaces. For example the first theorem of this type</p> <p>Brown, R. and Higgins, P.~J. On the connection between the second relative homotopy groups of some related spaces. Proc. London Math. Soc. (3) \textbf{36}~(2) (1978) 193--212. </p> <p>said that the fundamental crossed module functor from pairs of pointed spaces to crossed modules preserves certain colimits. This allows some calculations of homotopy 2-types and then you need further work to compute the 1st and 2nd homotopy group; of course these two homotopy groups are pale shadows of the 2-type. As example calculations I mention </p> <p>R. Brown Coproducts of crossed $P$-modules: applications to second homotopy groups and to the homology of groups'', {\em Topology} 23 (1984) 337-345.</p> <p>(with C.D.WENSLEY), Computation and homotopical applications of induced crossed modules', J. Symbolic Computation 35 (2003) 59-72.</p> <p>In the second paper some computational group theory is used to compute the 2-type and so 2nd homotopy groups of some mapping cones of maps $Bf: BG \to BH$ where $f:G \to H$ is a morphism of groups. </p> <p>For applications of the work with Loday I refer you for example to the bibliography on the nonabelian tensor product </p> <p>www.bangor.ac.uk/r.brown/nonabtens.html</p> <p>which has 100 items, and also </p> <p>Ellis, G.~J. and Mikhailov, R. A colimit of classifying spaces. {Advances in Math.} (2010) arXiv: [math.GR] 0804.3581v1 1--16.</p> <p>So in the tensor product work, we determine $\pi_3 S(K(G,1))$ as the kernel of a morphism $\kappa: G \otimes G \to G$ (the commutator morphism!). </p> <p>These theorems have connectivity conditions which means they are restricted in their applications, and don not solve all problems! There is still some interest in computing homotopy types of some complexes which cannot otherwise be computed. It is also of interest that the calculations are generally nonabelian ones. So the aim is to make some aspects of higher homotopy theory more like the theory of the fundamental group(oid), which is why I coined the term higher dimensional group theory' as indicating new structures underlying homotopy theory. </p> <p>Even the 2-dimensional theorem on crossed modules seems little known or referred to! The proof is not so hard, but requires the notion of the homotopy double groupoid of a pair of pointed spaces. See also some recent presentations available on my preprint page. </p> <p>Further comment: 11:12 24 Sept.</p> <p>The HHSvKT's have two roles. One is to allow some calculations and understanding not previously possible. People concentrate on the homotopy groups of spheres but what about the homotopy types of more general complexes? One aim is to give another weapon in the armoury of algebraic topology. </p> <p>The crossed complex work applies nicely to filtered spaces. The new book (pdf downloadable) gives an account of algebraic topology on the border between homotopy and homology without using singular or simplicial homology, and allows for some calculations for example of homotopy classes of maps in the non simply connected case. It gets some homotopy groups as modules over the fundamental group. </p> <p>I like the fact that the Relative Hurewicz Theorem is a consequence of a HHSvKT, and this suggested a triadic Hurewicz Theorem which is one consequence of the work with Loday. Another is determination of the critical group in the Barratt-Whitehead n-ad connectivity theorem - to get the result needs the apparatus of cat^n-groups and crossed n-cubes of groups (Ellis/Steiner). </p> <p>The hope (expectation?) is also that these techniques will allow new developments in related fields - see for example work of Faria Martins and Picken in diff geom. </p> <p>Developments in algebraic topology have had over the decades wide implications, eventually in algebraic number theory. People could start by trying to understand and apply the 2-dim HHSvKT! </p> http://mathoverflow.net/questions/39818/the-higher-van-kampen-theorems-and-computation-of-the-unstable-homotopy-groups-of/39871#39871 Answer by Carl Futia for The higher Van Kampen Theorems and computation of the unstable homotopy groups of spheres Carl Futia 2010-09-24T16:05:20Z 2010-09-24T16:05:20Z <p>I think it is important to remember that the Brown-Loday theorem concerns colimits of cat-n groups obtained from an n stage filtration of the underlying space. Moreover, cat-n groups can only provide information on the n-type.</p> <p>So, if you wanted to compute pi_200 of the two sphere, you would need a cat-200 group (at least). And if you wanted to apply the Brown_Loday VK theorem you would need a 200 stage topological filtration of the two sphere. Any thoughts ? or do I have this wrong?</p>