Abstract In this chapter we briefly introduce important concepts from homology theory and highlight how some results Chap. 3 can be understood through the lens of algebraic topology. This accumulates in showing that Euler characteristic by means CW complexes, or intersection theory.

Abstract This chapter presents the main general topological tools, as used in later parts. In particular we highlight manifolds, homotopies and retractions.

Abstract In this chapter we discuss differentiable structures on topological manifolds. particular, transversality, tubular neighbourhoods, index theory, the degree and theorems by Poincaré Hopf Bobylev Krasnosel’skiĭ.