Homology, Morse relations
In Chapter 2, 3, 4 we considered changes in the structure of lower level sets of functions up to homotopy equivalence (cell-attaching etc.). Consequently, we may measure those changes by means of homotopy invariants. In Chapter 1, Fig. 1.6.3, we have given already an intuitive impression of what may happen if we attach a 2-cell, in terms of canceling, respectively creating a 2-, respectively 3-dimensional “hole”. Now we will make the notion of a “k-dimensional hole” mathematically precise. In fact, given a topological Hausdorff space X, for q = 0, 1, 2, ... , we associate with X a vector space (over ℝ): H q (X). In case H q (X) is finite dimensional, the dimension r q of H q (X) will represent the number of “(q + 1)-dimensional holes” in X. In this section we will give a brief introduction to the theory on the spaces H q (X). As far as proofs are concerned, we restrict ourselves to those proofs which are important for a good understanding. For those proofs which we delete, we refer to [A/B], [Gre], [Span].
KeywordsExact Sequence Canonical Projection Hausdorff Space Topological Hausdorff Space Singular Homology
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