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  • Print publication year: 2010
  • Online publication date: May 2010

Introduction to unstable homotopy theory

Summary

Computation of the homotopy groups πn(X) of a topological space X has played a central role in homotopy theory. And knowledge of these homotopy groups has inherent use and interest. Furthermore, the development of techniques to compute these groups has proven useful in many other contexts.

The study of homotopy groups falls into three parts.

First, there is the computation of specific homotopy groups πn(X) of spaces. This may be traced back to Poincaré [106] in the case n = 1:

Poincaré: π1(X)/[π1(X), π1(X)] is isomorphic to H1(X).

Hurewicz [62] showed that, in the simply connected case, the Hurewicz homomorphism provides an isomorphism of the first nonzero πn(X) with the homology group Hn(X) with n ≥ 1:

Hurewicz: If X is an n − 1 connected space with n ≥ 2, then πn(X) is isomorphic to Hn(X).

Hopf [58] discovered the remarkable fact that homotopy groups could be nonzero in dimensions higher than those of nonvanishing homology groups. He did this by using linking numbers but the modern way is to use the long exact sequence of the Hopf fibration sequence S1S3S2.

Hopf: π3 (S2) is isomorphic to the additive group of integers ℤ.

Computation enters the modern era with the work of Serre [116, 118] on the low dimensional homotopy groups of spheres.

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