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Homology, Morse relations

  • Hubertus Th. Jongen
  • Peter Jonker
  • Frank Twilt
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 47)

Abstract

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].

Keywords

Exact Sequence Canonical Projection Hausdorff Space Topological Hausdorff Space Singular Homology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Hubertus Th. Jongen
    • 1
  • Peter Jonker
    • 2
  • Frank Twilt
    • 2
  1. 1.Department of MathematicsAachen University of TechnologyAachenGermany
  2. 2.Department of Mathematical SciencesUniversity of TwenteEnschedeThe Netherlands

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