Topology/Mayer-Vietoris Sequence

Topology Mayer-Vietoris Sequence Relative Homology Eilenburg-Steenrod Axioms A Mayer-Vietoris Sequence is a powerful tool used in finding Homology groups for spaces that can be expressed as the unions of simpler spaces from the perspective of Homology theory. Definition If X is a topological space covered by the interiors of two subspaces A and B, then is an exact sequence where . There is a slight adaptation for the reduced homology where the sequence ends instead Examples Consider the cover of formed by 2-discs A and B in the figure. thumb|250px|right| covered by 2-discs A and B The space is homotopy equivalent to the circle. We know that the homology groups are preserved by homotopy and so for and . Also note how the homology groups of A and B are trivial since they are both contractable. So we know that This means that since is an isomorphism by exactness. Consider the cover of the torus by 2 open ended cylinders A and B. thumb|280px|right|How we choose A and B. Exercises (under construction) Topology Mayer-Vietoris Sequence Relative Homology Eilenburg-Steenrod Axioms