I am studying nth relative homotopy groups from Hather.For a pair (X,A) where A$\subset$X nth-relative homotopy groups is defined by homotopy class of maps$(I^n,\delta I^n,J^{n-1})$ $\rightarrow$(X,A,$x_0$) where homotopies through these types of maps.

My question is how to calculate relative homotopy groups . I know that for a pair(X,A) fits into long exact sequence of homotopy groups.From these we can calculate relative homotopy group in some cases. Are there any method to calculate relative homotopy groups? Could anyone please explain this through some good examples? Any help would be appreciated.

  • 2
    $\begingroup$ What kind of answer would you expect? Computing homotopy groups is hard, computing relative homotopy groups is hard too. Unless you manage to find an ad-hoc argument, the long exact sequence is pretty much the only way of doing anything. $\endgroup$ Jul 15 '15 at 11:08
  • $\begingroup$ okkk...what do you mean by "ad-hoc argument"?Could you please explain that..thanks... $\endgroup$
    – Ripan Saha
    Jul 15 '15 at 11:23
  • $\begingroup$ something that is geared towards a specific situation $\endgroup$
    – Thomas Rot
    Oct 30 '15 at 13:29
  • $\begingroup$ @Najib Idrissi Just to point out that the work mentioned in my answer shows there are non ad hoc ways of dealing with relative homotopy groups. For more background, see presentations on my preprint page pages.bangor.,ac.uk/~mas010/brownpr.html, e.g. Aveiro, Galway. $\endgroup$ Jan 19 '16 at 17:21

There is a functor $\Pi_2$ which takes a based pairs of spaces $(X,A,x)$ to the crossed module $$\delta: \pi_2(X,A,x) \to \pi_1(A,x).$$ It seems not at all well known that this functor satisfies a Seifert-van Kampen type theorem, though the result was published in 1978 (Brown and Higgins, Proc. LMS); the result was inspired by, and has as a corollary, a theorem of JHC Whitehead in "Combinatorial Homotopy II" on free crossed modules, which in texts is sometimes stated, but rarely proved. Further applications were developed over the years and are explained in Part I of the book partially titled Nonabelian Algebraic Topology (EMS Tract in Math vol 15, (2011), pdf available). Naturally there are some connectivity conditions, which means the result calculates only some examples of second relative homotopy groups, or, one can say, only some homotopy 2-types.

The proof of this theorem involves double groupoids!

Part II of the above book discusses the higher dimensional case. So one gets the Relative Hurewicz Theorem as a Corollary of a Higher Homotopy Seifert-van Kampen Theorem.

19 Jan 2016 Just to clarify a bit: the results involve computations using pushouts of crossed modules. Analogously to the notion of fundamental groupoid on a set of base points, one considers "invariants" with structure in a range of dimensions, but the invariants are defined not on spaces but spaces with structure. So the general situation, see this presentation, is given by homotopically defined functors $$\mathbb H: (\text{Topological Data}) \to (\text{Algebraic Data})$$ which preserve certain colimits.

19 May, 2019 Actually my answer to this stackexchengwe question is relevant.


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