Holmstrom Homotopy groups

We can define the homotopy groups of a fibrant simplicial set, following Hovey chapter 3.4.

Let $x,y$ be 0-simplices in a fibrant simplicial set $X$. Say that $x$ is homotopic to $y$ if there is a one-simplex $z$ such that $d_0 z = x$ and $d_1 z = y$. This is an equivalence relation, and the set of equivalence classes is denoted $\pi_0 X$. This set is isomorphic to $\pi_0$ of the geometric realization. This construction is a functor from fibrant simplicial sets to sets.

For the other $\pi_k$, see Hovey page 85.

Goerss and Schemmerhorn: One of the lessons of the last thirty years is that in order to compute homotopy classes of maps, the best strategy can be to compute the homotopy type of the mapping space, and then read off the components (p. 27). Related: The hammock localization of a category with WEs.

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Homotopy groups

AT

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Homotopy groups

Serre: Bourbaki exp 44: Homotopy groups

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Homotopy groups

A PI-algebra is the algebraic structure looking like the homotopy groups of a space, roughly. This means it is a graded group with Whitehead products, compositions, and action of the fundamental group. See papers by Dwyer. Can define Quillen homology and cohomology of such a thing.

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Homotopy groups

Curtis: Some relations between homotopy and homology, Ann of Math 1965, showed that “the homotopy groups of a finite, simply connected simplicial complex are finitely computable”.

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Homotopy groups

Serre’s theorem on finiteness.

Ravenel mentions early in his orange book that under quite general hyps, homotopy classes of maps between two spaces form a countable set.

nLab page on Homotopy groups