# Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated?

Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated? Or are there some tables or handbooks which list some common calculated results of differential geometry, differential topology and algebraic topology?

• Most likely not. Why would one need such a table? And what is "every manifold"? – Alex Degtyarev Nov 25 '14 at 10:21
• @AlexDegtyarev I mean some common manifolds, such as SU(n), SO(n), RP^n and so on. Not every textbook of algebraic topology will calculate these all. If I will use some of them, I have to calculate them. Are there something like table of integrals which I can only consult when I need them. – 346699 Nov 25 '14 at 10:28
• Just don't use lousy textbooks. In my opinion, the ultimate textbook is Fomenko, Fuchs, "A course in homotopy topology". It contains all groups one can imagine! Alas, it's in Russian; I'm surprised that no one has translated it yet. – Alex Degtyarev Nov 25 '14 at 10:33
• math.columbia.edu/~khovanov/gradat2014/FFG1.pdf – bananastack Nov 25 '14 at 12:22
• I am very fond of Fomenko-Fuchs's book, but this is an introductory text and not a replacement for say Toda's book on homotopy groups of spheres. – Igor Belegradek Nov 25 '14 at 13:56

## 2 Answers

Matthias Kreck and collaborators have set out to create an encyclopaedic catalog of manifolds and their properties a while ago (http://www.map.mpim-bonn.mpg.de/Main_Page), although I don't think it's quite at a stage where one might describe it as useful. Its aim is to collect all known information about "real-world" manifolds, whatever that means. It's a wiki, so if you're interested in helping assemble such a catalog, you might consider contributing. Personally I think that the world of manifolds is too rich and, well, manifold, to be usefully catalogued in this way, but I'd love to be corrected.

Such tables indeed exist even though only partial information is available, of course. Higher homotopy groups of $RP^n$ are the same as those of spheres, so look at tables here. For Lie groups there is a good survey of Mimura ["Homotopy theory of Lie groups". Handbook of algebraic topology, 1995] with some tables if memory serves. There are also tables for exceptional Lie groups. If you want a more basic info there is a monograph "Topology of Lie groups I, II" by Mimura and Toda.

In general, I recommend a mathscinet search on "homotopy" and "Lie group" in the title. A number of things has happened since the above-mentioned works, and other information is in paper (not table) form. See e.g. here.

• Mimura's book also contains cohomology groups of homogeneous spaces. – user43326 Nov 25 '14 at 14:33