Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible to pass to Hamiltonian formulation, because this transform allows the construction of the cotangent bundle which is a sympletic manifold. Lagrangian formulation is a sub-set of Hamiltonian formulation, […]

Let G be a finitely generated abelian group and M a compact manifold, I want to prove that $H_r(M,G)$ is finitely generated for $r\ge 0 $. First I was thinking if I could do induction over $r$ because for $r=0$, $H_0(M,G) \cong \bigoplus_{i=1}^nG$ where $n$ denotes the number of connected components of M.But for $r>0$ […]

What are some examples of manifolds that do not have boundaries and are not boundaries of higher dimensional manifolds? Is any $n$-dimensional closed manifold a boundary of some $(n+1)$-dimensional manifold?

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