4 HOMOTOPY GROUPS

The idea of homology groups in the previous chapter was to assign a group structure to cycles that are not boundaries. In homotopy groups, however, we are interested in continuous deformation of maps one to another. Let $X$ and $Y$ be topological spaces and let $\mathcal{F}$ be the set of continuous maps, from $X$ to $Y$ . We introduce an equivalence relation, called ‘homotopic to’, in $\mathcal{F}$ by which two maps $f,g\in \mathcal{F}$ are identified if the image $f(X)$ is continuously deformed to $g(X)$ in $Y$ . We choose $X$ to be some standard topological spaces whose structures are well known. For example, we may take the $n$ -sphere $S^n$ as the standard space and study all the maps from $S^n$ to $Y$ to see how these maps are classified according to homotopic equivalence. This is the basic idea of homotopy groups.

We will restrict ourselves to an elementary study of homotopy groups, which is sufficient for the later discussion. Nash and Sen (1983) and Croom (1978) complement this chapter.

Problems

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

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