For questions regarding the structure and properties of compact manifolds.

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6
votes
2answers
552 views

Finitely generated singular homology

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 $r$ ...
10
votes
0answers
633 views

Homology of a compact manifold is finitely generated [duplicate]

Perhaps this is obvious and I am overlooking something, but why are the homology groups of compact manifolds finitely generated?
5
votes
2answers
234 views

How to Show Cotangent Bundles Are Not Compact Manifolds?

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 ...
1
vote
1answer
81 views

Partition of unity. Does this one exist?

Let $X:=\mathbb{R^n}$ be given and $M \subset X$ be a compact set in it. Then my question is: Are there $\alpha_i \in C^{\infty}(X,\mathbb{R})$ such that $supp(\alpha_i) \subset N$, where $N$ is an ...