I’m having trouble to prove the following statement: If the $n$-sphere $S^n\subset\mathbb{R}^{n+1}$ admits a (continuous) nonvanishing tangent vector field, then $n$ is odd. The idea of the proof is pretty clear on my mind: we use the fact that the antipodal map has degree $(-1)^{n+1}$ and that the Brouwer degree is invariant under homotopies. We […]

I hear there is a semi-famous theorem from my advisor, but he didn’t know the name and I was unable to find it online. Does anybody know of the following? Let $S$ be a compact Riemann surface. Then for a given genus $g$, up to isomorphism (holomorphism in our case), there are only finitely many […]

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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