Book contents
- Frontmatter
- Contents
- Preface to the third edition
- Preface to the first edition
- List of notation
- Introduction
- 1 Graphs
- 2 Closed surfaces
- 3 Simplicial complexes
- 4 Homology groups
- 5 The question of invariance
- 6 Some general theorems
- 7 Two more general theorems
- 8 Homology modulo 2
- 9 Graphs in surfaces
- Appendix: abelian groups
- References
- Index
5 - The question of invariance
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to the third edition
- Preface to the first edition
- List of notation
- Introduction
- 1 Graphs
- 2 Closed surfaces
- 3 Simplicial complexes
- 4 Homology groups
- 5 The question of invariance
- 6 Some general theorems
- 7 Two more general theorems
- 8 Homology modulo 2
- 9 Graphs in surfaces
- Appendix: abelian groups
- References
- Index
Summary
This chapter has a carefully chosen title: we are concerned here primarily with the question, and not with the answer. The question is this: suppose that K and K1, are simplicial complexes triangulating the same object, i.e. with ∣K∣ = ∣K1∣. Is it true that Hp(K) ≅ Hp(K1) for all p? The answer is in fact ‘yes’ – indeed a stronger assertion (5.13) is true, namely that the isomorphisms hold if ∣K∣ is merely supposed homeomorphic to ∣K1∣, i.e. if there exists a continuous and bijective map f: ∣K∣ → ∣K1∣. (The inverse map f−1 will automatically be continuous by a theorem of general topology, using the ‘compactness’ of ∣K∣. See the books of Kelley or Lawson listed in the References.) The stronger assertion is referred to as the topological invariance of homology groups: it says that homology groups do not depend on particular triangulations but only on the underlying topological structure of ∣K∣.
The topological invariance result can be restated in a negative form: if there exists an integer p such that Hp(K) and Hp(K1) are not isomorphic, then ∣K∣ and ∣K1∣ are not homeomorphic. Using results of Chapter 6 this in turn implies that no two inequivalent closed surfaces (2.8) have homeomorphic underlying spaces.
But we shall not prove the topological invariance theorem 5.13, since it would divert attention from the developments of succeeding chapters. The only invariance result actually needed is a much weaker one, and that one is proved in full; see 5.7.
- Type
- Chapter
- Information
- Graphs, Surfaces and Homology , pp. 127 - 137Publisher: Cambridge University PressPrint publication year: 2010