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Erschienen in:
2013 | OriginalPaper | Buchkapitel
6. Obstruction Theory
verfasst von : Phillip Griffiths, John Morgan
Erschienen in: Rational Homotopy Theory and Differential Forms
Verlag: Springer New York
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Abstract
Recall, in the proof of the Whitehead theorem, we showed that if X is a CW complex and Y is a space with \(\pi _{i}(\mathrm{Y}) = 0\) for all i ≥ 0, then any map \(\mathrm{f}: \mathrm{X} \rightarrow \mathrm{ Y}\) is homotopic to the base point map \(\mathrm{X} \rightarrow \mathrm{ y}_{0} \in \mathrm{ Y}\). The proof was by induction over the skeleta of X. Obstruction theory is a generalization of this technique to the case when the homotopy groups of Y are not necessarily zero. It does not give a complete understanding of when a map \(\mathrm{f}: \mathrm{X} \rightarrow \mathrm{ Y}\) is homotopic to a constant, or more generally when two maps \(\mathrm{f}_{1},\mathrm{f}_{2}: \mathrm{X} \rightarrow \mathrm{ Y}\) are homotopic, but it gives some insight.