To give an introduction to the topology of manifolds, emphasising the role of homology as an invariant and the role of Morse theory as a visualising and calculational tool.

(LO1) To be able to:
•   give examples of manifolds, particularly in low dimensions;
•   compute homology groups, Euler characteristics and degrees of maps in simple cases;
•   determine whether an explicitly given function is Morse and to identify its critical points and their indices;
•   use the Morse inequalities to estimate the ranks of homology groups;
•   use the Morse complex to compute Euler characteristics and, in simple cases, homology.

Smooth manifolds (embedded in R n ): coordinate charts, tangent spaces, examples, manifolds with boundary. Homology: chains, cycles, boundaries, the definition of homology and simple examples. Functoriality of homology, homotopy equivalences and the homotopy invariance of homology. Exact sequences of Abelian groups. Relative homology and the long exact sequence of a pair (proof non-examinable). Excision (statement only). Computations of homology. Applications to degrees of maps, Euler characteristics and fixed point theorems. Morse theory: smooth maps, non-degenerate critical points, the Hessian and the index. Morse functions, Sard’s lemma and the existence of Morse functions (non-examinable), the Morse lemma. Change in homotopy type at critical values and consequences for homology. The Morse inequalities. Brief treatment of the Morse complex. Applications to Euler characteristics, computations of homology and Poincare duality.