Chapter 4 Tools from Algebra and Topology

This chapter is a digression in algebra and topology. We saw earlier that the tangent space construction gives rise to a functor from the category of C ∞ manifolds (M, p) with a marked point to the category of vector spaces. Much of differential topology and differential geometry consists of trying to see how much of the geometric information is encoded in the linear algebra of tangent spaces. To this end, we need a larger arsenal of algebraic techniques than linear spaces and linear maps.