Algebraic Topology: Homology and Cohomology
This self-contained text is suitable for advanced undergraduate and graduate students and may be used either after or concurrently with courses in general topology and algebra. It surveys several algebraic invariants: the fundamental group, singular and Cech homology groups, and a variety of cohomology groups.
Proceeding from the view of topology as a form of geometry, Wallace emphasizes geometrical motivations and interpretations. Once beyond the singular homology groups, however, the author advances an understanding of the subject's algebraic patterns, leaving geometry aside in order to study these patterns as pure algebra. Numerous exercises appear throughout the text. In addition to developing students' thinking in terms of algebraic topology, the exercises also unify the text, since many of them feature results that appear in later expositions. Extensive appendixes offer helpful reviews of background material.
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algebraic homotopy inverse boundary operator called Cech homology groups cell decomposition chain complex chain groups chain homomorphism coboundary cochain cocycle coeﬂicient group coﬁnal cohomology class constructed contained continuous map coordinates corresponding deﬁned Deﬁnition denoted dimension direct sum dual element example excision theorem Exercise f and g ﬁnite open covering ﬁrst following diagram formula geometric Hence homo homology sequence homology theory homomorphism f homotopy theorem Hp(E identiﬁed identity map implies inclusion map induced homomorphisms integers intersection inverse limit inverse system isomorphism kernel Lemma Let f Let G linear map map f morphism notation Note obtained open set oriented cell oriented simplexes oriented simplicial pair of spaces Proof proved reﬁnement relation represented respectively result satisﬁed shown simplicial chain simplicial complex simplicial homology simplicial map simplicial pair singular chain singular homology singular simplex subcomplex subset subspace system of homomorphisms topological space triangulable vertexes zero