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Edit Jan 11, 2014 Further to my last point, consider my answer on excision for relative $\pi_2$: http://math.stackexchange.com/questions/617018/failure-of-excision-for-pi-2/621723#621723

Edit Jan 11, 2014 Further to my last point, consider my answer on excision for relative $\pi_2$: https://math.stackexchange.com/questions/617018/failure-of-excision-for-pi-2/621723#621723

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

In the second paper some computational group theory is used to compute the 2-type, and so 2nd homotopy groups as modules, for some mapping cones of maps $ Bf: BG \to BH$ where $f:G \to H$ is a morphism of groups.

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.

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.

December 28, 2015 I mention also a presentation at CT2015 Aveiro on A philosophy of modelling and computing homotopy types. Note that homotopy groups are but a pale "shadow on a wall" of a homotopy type. Note also that homotopy groups are defined only for a space with base point, i.e. a space with some structure. My work with Higgins and with Loday involves spaces with much more structure; that with Loday involves $n$-cubes of pointed spaces. As with any method, it is important to be aware of what it does, and what it does not, do. One aspect is that the work with Loday deals with nonabelian algebraic models, and obtains when it applies precise colimit results in higher homotopy. One inspiration was a 1949 Theorem of JHC Whitehead in "Combinatorial Homotopy II" on free crossed modules.

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

In the second paper some computational group theory is used to compute the 2-type, and so 2nd homotopy groups as modules, for some mapping cones of maps $ Bf: BG \to BH$ where $f:G \to H$ is a morphism of groups.

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 armory of algebraic topology.

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 differential geometry.

December 28, 2015 I mention also a presentation at CT2015 Aveiro on A philosophy of modelling and computing homotopy types. Note that homotopy groups are but a pale "shadow on a wall" of a homotopy type. Note also that homotopy groups are defined only for a space with base point, i.e. a space with some structure. My work with Higgins and with Loday involves spaces with much more structure; that with Loday involves $n$-cubes of pointed spaces. As with any method, it is important to be aware of what it does, and what it does not, do. One aspect is that the work with Loday deals with nonabelian algebraic models, and obtains, when it applies, precise colimit results in higher homotopy. One inspiration was a 1949 Theorem of JHC Whitehead in "Combinatorial Homotopy II" on free crossed modules.

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.

which has 130 items (Jan, 2014), since the topic has been taken up by group theorists, and also

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!). Edit: In fact we compute the 3-type of $SK(G,1)$ so you can work out the Whitehead product $\pi_2 \times \pi_2 \to \pi_3$ and composition with the Hopf map $\pi_2 \to \pi_3$.

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.

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.

In the second paper some computational group theory is used to compute the 2-type, and so 2nd homotopy groups as modules, for some mapping cones of maps $ Bf: BG \to BH$ where $f:G \to H$ is a morphism of groups.

which has 144 items (Dec. 2015: the topic has been taken up by group theorists, because of the relation to commutators) and also

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!). In fact we compute the 3-type of $SK(G,1)$ so you can work out the Whitehead product $\pi_2 \times \pi_2 \to \pi_3$ and composition with the Hopf map $\pi_2 \to \pi_3$.

So the aim is to make some aspects of higher homotopy theory more like the theory of the fundamental group(oid); this is why I coined the term `higher dimensional group theory' as indicating new structures underlying homotopy theory.

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.