# Questions tagged [compact-manifolds]

For questions regarding the structure and properties of compact manifolds.

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### A regular, connected, compact surface with curvature on $[0,1]$

today was my final differential geometry exam and there was a problem that I partially solved, but I have some doubts. The problem asked to prove that there exists a regular, connected, compact ...
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### Does every finitely presented group have a finite index subgroup with free abelianisation?

Let $G$ be a finitely presented group. Does there exist a finite index subgroup $H$ such that its abelianisation $H^{\text{ab}} = H/[H, H]$ is free abelian? Note, if $G^{\text{ab}}$ is not already ...
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### $Ric$ flat metric on $S^n$.

Suppose $\mathbb{R}\times S^{n}$ admits a complete riemannian metric $g$ such that $Ric_g = 0$. Prove that this metric $g$ induces a metric $\tilde g$ on $S^{n}$ such that $Rig_{\tilde g} = 0$. So far,...
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### Handles have the form $D^λ×D^{m−λ}$

I'm studying Matsumoto's An Introduction to Morse Theory. I want to solve a problem on page 76. Context: Let $M$ be a closed $m-$manifold and $f:M\rightarrow \mathbb{R}$ a Morse function. Let $c$ be a ...
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### U(n) is compact and algebraic, but not abelian—why not a contradiction?

the subgroup of unitary matrices $\text{U}(n) \subset GL(n, \mathbb{C})$ is compact and definitely algebraic, with an algebraic group law; on the other hand, it's not abelian. why is this not a ...
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### Lie subgroup of non-abelian compact Lie group is compact?

I need to decide if this statement is true or false: " Every Lie subgroup of non-abelian compact Lie group is compact." I think that it is false. I thought in a counterexample in which the ...
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### Proof Concerning homeomorphisms of $\mathbb{P}^2$

Is the following proof valid? CLAIM: The space obtained by attaching a disc to a Mobius Strip along the boundary is homeomorphic to the projective plane. PROOF: We begin by showing that the boundary ...
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### Prove that every compact manifold is homeomorphic to a subset of some Euclidean space.

I am trying to prove the following theorem: Theorem. Every compact manifold is homeomorphic to a subset of some Euclidean space. The manifolds I'm considering are the most general (without any ...
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### Could exists a vector field on $\mathbb{S}^{2}$ with exactly $n$ zeroes?

I just started to learn index theory of tangent vector fields. I'm aware of two examples on the sphere $\mathbb{S}^{2}$ with exactly one zero, which, which are $F(x,y) = (1-x^2-y^2)\partial x$ thought ...
306 views

### Compact hyperbolic three-manifold - a question

In a recent paper the authors considered a spacetime described by $$AdS_4 \times \Sigma_3 \times \mathcal{I}_r \times S^2$$ where $\mathcal{I}_r$ is an interval of the $r$-coordinate and the two-... 397 views

### Homotopy invariance for compactly supported cohomology

I can't find any reference regarding homotopy invariance for compactly supported cohomology and I wonder under which conditions the homotopy invariance still holds for compactly supported cohomology. ...
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### Uniformization theorem for $C^k$ surfaces?

Does the uniformization theorem apply for surfaces that are $C^k$ ($k<\infty$)? I'm familiar with a couple of proofs of Uniformization (using Riemann-Roch, Ricci flow). But most of these proofs ...
1 vote
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### Is there an example of reducible compact 3-manifold with boundary that has no embedded incompressible two-sided surface?

There is a theorem stating that for irreducible compact manifolds with non-empty boundary there always exists such an embedded surface and I'm trying to understand why the irreducibility condition ...
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1 vote
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### Compactification of log z Riemann Surface

I've been reading the 'Road to Reality' book of Roger Penrose and in the chapter on Riemann Surfaces, there is a note that we can compactify the log z Riemann Surface into a sphere. But I don't see ...
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### What is the "natural homomorphism" in the definition of an *essential manifold*?

The following definition of "essential manifold" is in this wiki page: A closed $n$-manifold $M$ is called essential if its fundamental class $[M]$ defines a nonzero element in the homology ...
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### Simplicial complexes embedded on a compact manifold

Every finite graph can be embedded on some compact surface of sufficient genus such that no two edges cross. If $S$ is a finite simplicial complex of dimension $n$, can $S$ be embedded in some ...
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### Can all connected graphs be embedded on a closed, compact 2-Manifold?

I know that there are spherical (planar) graphs such as $K_4$,and toroidal graphs such as $k_7$, but I was wondering if given any connected graph $G$, there exists a closed, compact 2-Manifold $M$ ...
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### An orientable surface that cannot be embedded into $\Bbb R^3$?

By Whitney's embedding theorem, every 2-dimensional manifold (aka. a surface) can be embedded into $\Bbb R^4$. Now, the Wikipedia page on that theorem states in this paragraph that we can even embedd ...
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### intersection number on the boundary of a manifold

Let $F: W \to N$ be a smooth map, where $W$ is a compact manifold with boundary, $Z \subset N$ is closed and all manifolds are oriented. Also $\partial F \pitchfork Z$ and $F^{-1}(Z)$ is a compact, ...
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### Proof of classification theorem for compact surfaces

I am reading Massey's 'A basic coruse in Algebraic Topology'. In first chapter, he proved classification theorem for compact surfaces (compact connected 2-manifold). This theorem classifies compact ...
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### Can one characterize compact smooth manifolds dynamically as such?

If $M$ is a smooth (connected) compact manifold, then it is known that any (smooth) vector field is complete, which means that the flow exists for all $t \in \mathbb{R}$. What about the converse? If \$...