General Topology and Homotopy Theory - ISBN:3540909702

General Topology and Homotopy Theory

Date: 21 April 2011, 05:00

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In this monograph, based on a set of sixteen lectures to students, the author expounds certain parts of general topology which are particulary relevant to homotopy
theory. His book is reasonably self-contained. It is tightly written and elegant.
Some idea of the book's content may be gathered from the titles of its eight chapters: The basic framework, The axioms of topology, Spaces under and spaces over,
Topological transformation groups, The notion of homotopy, Cofibrations and fibrations, Numerable coverings, Extensors and neighbourhood extensors.
Another way of summarizing the book is to paraphrase the seven propositions which the author labels ``theorems'', since these are clearly the highlights. He classifies
the numerable $G$-bundles for any topological group $G$ (Theorem 7.52). He proves under various hypotheses that a fiber-preserving map which is locally a fiber homotopy
equivalence is globally so (Theorems 7.57, 8.39, 8.40). Under suitable hypotheses, he relates the Lyusternik\mhy Shnirel?man category of a space $X$ to the
nilpotency class of the (nilpotent) group of homotopy classes of maps from $X$ to a path-connected topological group or grouplike space (Theorems 8.15, 8.16). When $G$
is a compact Lie group, when $B$ is a paracompact $G$-space, when $E$ is a neighbourhood $G$-extensor over $B$, and when $E^H$ is an extensor over $B^H$ for all closed
subgroups $H$ of $G$, he proves that $E$ is a $G$-extensor over $B$ (Theorem 8.49). These major theorems do not involve algebraic computations; rather they belong to
that part of general topology which uses open covers, quotients, fiber spaces and group actions. To set them up properly, much must be done, and that is the content of
the book.

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