# Tagged Questions

For questions regarding the structure and properties of compact manifolds.

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### Bounding a Sobolev norm on product manifold

Suppose you have a closed manifold $M$ and a function $f:M\times M\rightarrow\mathbb{C}$ that is $C^2$ in the $M$ variables, and suppose that at any point $q\in M$, we have $$f(\cdot,q)\in H_m(M).$$ ...
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### Suppose $M$ has trivial 1-st de Rham cohomology group. For which integers $k$ does there exist a smooth map $f : M → T^n$ of degree $k$?

Let $M$ be a compact oriented smooth $n$-manifold, with $H_{dR}^1(M)=0$. For which integers $k$ does there exist a smooth map $f : M → T^n$ of degree $k$? I know that if $M$ is simply-connected, we ...
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### integral constraint induce a manifold on Sobolev space

given the set $$M:=\{u\in H^2(\Omega):\int_{\Omega}u=m\,\}$$ $m\in \mathbb{R}$, $\Omega$ is a bounded piecewise smooth domain in $\mathbb{R}^n$. also denote by $u(t)$ a map: $u(t):(0,T)\to M$ ...
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### Finding the boundary of the continuous image of a compact simply-connected Lie group

Statement of the problem Given a continuous map $f:G \rightarrow D^2$ where $G$ is a compact simply connected Lie group and $D^2$ is the unit disk in the plane, I have shown that: There exists a ...
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### Hypersurfaces of a hemisphere

Let $S_+^{n+1}$ be the open hemisphere of the standard euclidean sphere centered at the north pole and let $M^n$ be a compact, connected and oriented hypersurface of $S_+^{n+1}$. Is it true that if $M$...
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### Surjectivity of the exponential map on SO(2n)/U(n)

Let $M:=SO(2n)/U(n)$ the homogeneous space of all orthogonal almost-complex structures on $\mathbb{R}^{2n}$. When $n=2$, it is known that $M$ is just the 2-sphere. 1) On the 2-sphere, the ...
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### Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$?

The first question is Is the hemisphere of $S^4$ the unique compact 4-dimensional manifold with $\partial K = S^3$? In other words, is it obvious? The question stems from a theoretical physics ...
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### How can I show $R^n$ is dense in $S^n$?

How can I show $R^n$ is dense in $S^n$? I wanted to show $S^n$ is compactification of $R^n$. for this I need $R^n$ is not compact, for this there is no problem, and $S^n$ is compact, I did it with ...
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### Prove or disprove: for every nonzero $\phi \in H^k(M),$ there is $\psi \in H^{n-k}(M)$ with $\phi \cup \psi \neq 0$

...for $M$ a compact, connected, orientable manifold without boundary. This is part of a question in a past paper I'm working on: the full thing is at https://www1.maths.ox.ac.uk/system/files/...
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### Total Gaussian curvature

For a compact surface, $S$, in $\mathbb{R}^3$, how would I go about showing that the total Gaussian curvature $\int_S K da \leq 4 \pi$? I feel like Hopf's Umlaufsatz and the Gauss-Bonnet Theorem are ...
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### line sub-bundles over an $n$ -sphere.

Could someone explain to me why are line bundles over any real $n$-sphere trivial bundles ( $n > 1$ ) ? Could someone tell me, in the case where $n = 2k$ with $k \in \mathbb{N}^*$ , why does ...
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### Is there a preferred way to characterize a cone in cohomology?

In a Euclidean space, one can of course describe a cone using a generalization of polar coordinates, and since the de Rham cohomology spaces are themselves Euclidean spaces, one can do the same here. ...
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### Are there any nontrivial second Hurewicz homomorphisms for familiar compact 6-dimensional manifolds?

Based on several computations I have done, it seems that the second Hurewicz homomorphism $$h:\pi_{2}(X)\rightarrow H_{2}(X;\mathbf{Z})$$ has a habit of being trivial. For instance, this seems to be ...
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### Lens Space Orientation Reversing Homeomorphism

I am thinking of an example where the connected sum of two three Manifolds depends on the chosen orientation. Hempel gives in his book "3-Manifolds" an example, namely lens spaces. He shows that two ...
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### Is log-general type an intrinsic property of a variety

Let $X$ be a smooth quasi-projective variety over $\mathbb C$. Let us say that $X$ is of log-general type if for some choice of smooth compactification $\bar X$ with normal crossings boundary divisor ...
Consider the manifold of constant negative curvature $G=SL(2, R)/\Gamma$ where $\Gamma$ is such that $G$ is compact (I have no special constraint on $\Gamma$). I know that by the Whitney embedding ...
For a smooth map $f:M\to N$ from an orientable closed surface $M$ to a non-orientable closed surface $N$, we define its parity (also called modulo 2 degree, and denoted $\deg_2(f)$) as the parity of ...