Nonlinear Analysis and Semilinear Elliptic Problems
With many illustrations and helpful exercises, this book covers a number of more modern topics from critical point theory to elliptic partial differential equations.
TABLE OF CONTENTS Nonlinear Analysis and Semilinear Elliptic Problems
Chapter 12: Morse Theory
This chapter is devoted to the basic theory developed by M. Morse, which relates the structure of the critical points of a regular function on a manifold to its topology.
12.1 A Short Review of Basic Facts in Algebraic Topology
In this section we recall some basic notions and results in algebraic topology. To keep the presentation short, we will introduce the concepts in an axiomatic way, referring the interested reader to more complete treatments, like the books [131, 162]. Then we will review briefly the explicit construction of the singular homology theory, omitting most of the proofs.
12.1.1 The axiomatic construction
Definition 12.1
Let (G i ) i be a sequence of Abelian groups, and let ( ? i ) i be a sequence of homomorphisms
We say that the sequence is exact if for every i there holds im ( ? i ) = ker ( ? i +1 ).
Example 12.2
(i) Let G 1, G 2 be Abelian groups, and consider the following part of a sequence
Then we have exactness if and only if ? is an isomorphism.
(ii) Let G 1, G 2, G 3 be Abelian groups, and suppose the following part of the sequence is exact
Then from the exactness it follows that ? 1 is injective, and that the image of ? 1 (which is isomorphic to G 1) is equal to the kernel of ? 2. Still...
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