Applications of Mayer-Vietoris

Algebraic Topology · Axiom Academy

LESSON Applications of Mayer-Vietoris Practical techniques for computing homology groups using the Mayer-Vietoris sequence 1. Computing Homology of Spheres We compute H n (S n ) by decomposing the n-sphere into two contractible hemispheres. The key is that their intersection is homotopy equivalent to S n-1 . Since U and V are contractible, H k (U) = H k (V) = 0 for k > 0. The M-V sequence gives us: This shows that H k (S n ) ≅ H k-1 (S n-1 ) for k > 0, giving us an inductive computation! The connected sum M#N is formed by removing a small ball from each manifold and gluing along the boundary. M-V allows us to relate H * (M#N) to H * (M) and H * (N). For closed orientable surfaces (n=2), this yields the crucial formula: 3. Applications to Surface Homology We can compute the homology of any closed orientable surface Σ g (genus g) by viewing it as a connected sum of g tori. Using the connected sum formula repeatedly: This gives us the complete homology groups of surfaces of any genus! The suspension ΣX of a space X is obtained by "suspending" X between two points. M-V proves the fundamental suspension isomorphism. Since both cones are contractible, the M-V sequence yields: The wedge sum X ∨ Y is formed by gluing X and Y at a single point. M-V can compute its reduced homology in terms of the reduced homology of X and Y. The M-V sequence, combined with the fact that U ≃ X and V ≃ Y while U ∩ V is contractible, gives:

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