The Mayer–Vietoris Sequence

Abstract

Although it is possible to compute homology directly from the definition, it is not always much fun to do so—computing the homology for a genus-g surface would require a lot of simplices and matrix manipulations! We were able to compute the fundamental group for an arbitrary surface using the Seifert–Van Kampen Theorem, breaking it up into smaller regions and splicing together their fundamental groups. In particular, we were able to express $\pi _1(A\cup B)$ in terms of $\pi _1(A)$, $\pi _1(B)$, $\pi _1(A\cap B)$, and some information about how they all fit together.