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.
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© 2013 Springer Science+Business Media New York
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Griffiths, P., Morgan, J. (2013). Obstruction Theory. In: Rational Homotopy Theory and Differential Forms. Progress in Mathematics, vol 16. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-8468-4_6
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DOI: https://doi.org/10.1007/978-1-4614-8468-4_6
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Publisher Name: Birkhäuser, New York, NY
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Online ISBN: 978-1-4614-8468-4
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