For questions regarding the structure and properties of compact manifolds.

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12
votes
1answer
110 views

If the connected sum of a manifold $M$ with itself gives back $M$, does it imply $M$ is a sphere?

Let $M$ be a compact, connected, oriented $n$-dimensional manifold without boundary. Suppose that $M\#M\cong M$. Does it imply that $M \cong S^n$? Sorry if this is a naive question. This is not my ...
11
votes
0answers
315 views

Norm Inequality on a Compact Riemannian Manifold

Consider the following problem: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality $$\Delta u \geq -...
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?
7
votes
2answers
346 views

Dimension of de Rham Cohomology groups?

Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
7
votes
2answers
568 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 $r$ ...
6
votes
1answer
369 views

Uniformization Theorem for compact surface

Why in proof of proposition 6 of http://arxiv.org/abs/0909.1665, they claim that if a embedded surfaces $\Sigma^2 \subset (M^3,g)$ is homeomorphic to $\mathbb{RP}^2$, where $M$ is compact manifold, ...
5
votes
3answers
409 views

Examples of manifolds that are not boundaries

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 ...
5
votes
2answers
236 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 ...
5
votes
4answers
237 views

Is $[0,1]$ a 1-manifold?

Is $[0,1]$ a 1-manifold? I would say no because at either endpoint the open sets containing it aren't homeomorphic to a 1-ball in $\mathbb R^1$.
4
votes
2answers
263 views

Why spheres are not symplectic manifolds?

Reading some books on diferential geometry, a found that $S^{2n}$ (with $ n>1$) are not symplectic manifolds. They say it's because the de Rham cohomology of this spheres are R, but I do not ...
4
votes
2answers
163 views

Is the additive group of real numbers (R,+) compact?

I have a very naive question about basic topology. My goal is to determine the conditions on a vector field in order for its flow to define a proper (R,+)-action. I understand that if the group G is ...
4
votes
2answers
68 views

connected sum of surfaces is well defined proof attempt

Suppose $S_1$ and $S_2$ are compact surfaces (connected 2-dimensional manifolds). If we cut out of them two closed disks, and glue the surfaces along disk boundaries we get new surface, their ...
4
votes
4answers
206 views

1-manifold is orientable

I am trying to classify all compact 1-manifolds. I believe I can do it once I can show every 1-manifold is orientable. I have tried to show prove this a bunch of ways, but I can't get anywhere. ...
4
votes
2answers
91 views

Do there exist cancellable manifolds?

I do not know whether there exists a terminology for that property, but let us say that a closed manifold $C$ is cancellable if for every closed manifolds $M_1$ and $M_2$, $C \times M_1$ and $C \times ...
4
votes
1answer
111 views

Moving a compact submanifold off of another submanifold?

This is an intuitive idea that I see referenced a lot. Consider the following situation. Let $M$ and $N$ be submanifolds, $M$ compact, in some larger manifold $X$. Suppose also that $\dim(M)+\dim(N)&...
4
votes
1answer
358 views

Schrödinger Kernels on manifolds

Let $M$ be a compact Riemannian manifold and $\Delta$ be the Laplace-Beltrami operator. It is well-known that the solution operator to the heat equation $e^{t \Delta}$ is smoothing for $t>0$ and ...
4
votes
1answer
145 views

Showing that $\mathbb S^1$ is a deformation retract of the Mobius strip, rigorously.

Intuitively, I can see why this is. I've found a few threads about this, but they only provide, for example, a deformation retraction of $I \times I$ to its diagonal $D = \{ (x,x) \in I \times I \}$, ...
4
votes
1answer
49 views

Finitely Many genus-g Quotients of Compact Riemann Surface

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 ...
4
votes
2answers
234 views

Integrals of Pullbacks

This is a problem from Guillemin's Differential Topology: Suppose that $f_0, f_1: X \to Y$ are homotopic maps and that the compact boundaryless manifold $X$ has dimension $k$. Prove that for all ...
4
votes
2answers
301 views

What is group manifold of a compact Lie Group?

I searched on google the meaning of a group manifold of a compact lie group but I didn't get the answer. A paper on arxiv "Background Independent Quantum Gravity:A Status Report- Abhay Ashtekar" on ...
4
votes
0answers
69 views

Map of smooth manifolds

Let $M$ and $N$ be smooth, connected $n$-dimensional manifolds. Let $M$ be compact and non-empty. Show that every embedding $f: M \to N$ is a diffeomorphism. So because $f$ is a embedding we have ...
3
votes
1answer
120 views

Approach topological manifolds with smooth manifolds

Because I'm doing some problems that consider all the manifolds while the situation is really clear when considering only smooth manifolds. Thus my question is can we always appoint a topological ...
3
votes
2answers
90 views

Algebraic surface as a smooth manifold

Let $S$ be the set of points $x=(x_1,x_2,\ldots,x_9)\in \mathbb{R}^9$ which satisfy the following conditions: $$x_1^{2}+x_2^{2}+x_3^{2}=x_4^{2}+x_5^{2}+x_6^{2}=x_7^{2}+x_8^{2}+x_9^{2}=1$$ $$x_1x_4+...
3
votes
1answer
47 views

Why is it impossible for a compact connected non-orientable $n$-manifold to be the suspension of some connected based space?

Suppose $M$ is a compact $n$-manifold (without boundary) that equals the reduced suspension $\Sigma Y$ of a connected based space $Y$. Why must $M$ be orientable? I am aware that cup products vanish ...
3
votes
1answer
169 views

A literature reference for Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth Riemannian manifolds

Anyone know a respectable reliable reference for the definition of Sobolev mappings $W^{m,p}(M,N)$ for M, N smooth compact Riemannian manifolds. It suffices for m natural and $p\geq 1$
3
votes
1answer
60 views

Volumes of hyperbolic manifolds

In a talk I attended the speaker said that the volume of a closed hyperbolic manifold is a topological invariant. Are known volumes rational? Irrational? Transcendental?
3
votes
2answers
35 views

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 ...
3
votes
1answer
166 views

Prove that there are no convex functions on compact manifolds

This one seems intuitively obvious to me but I don't know how to prove it. Suppose you have a compact manifold $M$ with a function $f$ defined on it. Given two points $x$ and $y$ on the manifold, ...
3
votes
2answers
574 views

A non orientable closed surface cannot be embedded into $\mathbb{R}^3$

Can someone please remind me how this goes? Here's the idea of proof I'm trying to recall: let $S$ be a closed surface (connected, compact, without boundary) embedded in $\mathbb{R}^3$. Then one can ...
3
votes
1answer
70 views

Momentum is quantised in compact spaces?

Background One of the first examples given when studying quantum mechanics is the particle on a cylinder, or particle on a ring. One finds that because of the periodic boundary conditions, ...
3
votes
1answer
245 views

Proof that a map from an orientable surface to a non-orientable surface has even degree.

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 ...
3
votes
0answers
84 views

Measurability of points regular

I'm reviewing the proof of the theorem of oseledet the book Mañe: Let $M$ a compact metric space and $f:M \rightarrow M$ a homeomorphism, $\pi: F \rightarrow M$ a finite-dimensional continuos vector ...
2
votes
1answer
36 views

How can uncountably many closed smooth 4-manifolds be presented by an essentially countable alphabet (Kirby diagrams)?

A smooth, closed 4-manifold admits a handle decomposition which is specified completely by its Kirby diagram. A Kirby diagram, up to isotopy, can be seen as a labelled morphism in the tangle category. ...
2
votes
1answer
113 views

a theorem in topology

Is there anyone know there is a theorem in topology which states that a compact manifold "parallelizable" with N smooth independent vector fields. must be an N-torus? and why the vector field here is ...
2
votes
1answer
20 views

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/...
2
votes
2answers
37 views

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 ...
2
votes
1answer
81 views

Can a product of a Stein manifold and a compact manifold be again Stein?

A Stein manifold is a manifold which is holomorphically separable and convex. It is well known that a product of two holomorphically convex (resp. Stein) manifolds is again holomorphically convex (...
2
votes
1answer
72 views

Equivalent definitions of Euler characteristic for closed manifolds

It is well-known that the Euler characteristic of a closed manifold $M^n$, which can be defined as $\chi(M)=\sum_{k=0}^n (-1)^k \operatorname{dim}H^k(M)$, equals the intersection number $I(\Delta,\...
2
votes
1answer
89 views

Why can an orientation on $X$ be written as a sum of cycles on this open cover?

I'm reading a proof of the following theorem Let $X$ and $Y$ be compact connected oriented $n$-manifolds, and let $f:X\to Y$ be continuous. Let $y\in Y$, and assume that $f^{-1}(y)$ is finite. ...
2
votes
1answer
156 views

Every compact orientable surface with $S^1\times\{0\}$ as its boundary intersects the $z$-axis

Let $M$ be a compact orientable surface (manifold in $\mathbb R^3$) with boundary $S^1\times\{0\}$.Show that $M$ intersects the $z$-axis. Some ideas: $1)$Since $M$ is a compact orientable manifold ...
2
votes
1answer
36 views

Coverings of a three-manifold

He guys, I have two questions regarding the following: Consider the three-manifold $\mathbf{T}^3 = S^1 \times S^1 \times S^1$ and let $S_n$ be the permutation group acting on $n$ letters. Let $\phi:\...
2
votes
0answers
40 views

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 ...
2
votes
0answers
38 views

A very general question about blow-up for experienced symplectic topologists and algebraic geometers

I am trying to understand the process of symplectic blow-up of compact symplectic manifolds $M^{2n}$ along compact symplectic submanifolds $X$ on a deeper level, and I am also searching for relevant ...
2
votes
0answers
96 views

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 ...
2
votes
1answer
106 views

orientation of symplectic manifold and lagrangian submanifolds

A statement: The self-intersection index of lagrangian submanifold $M \subset X$ is equal to Euler characteristic $\chi(M)$. How I should oriented $X$? Let's consider some example. The null-section $...
2
votes
0answers
81 views

Standards in P.L. Topology

About a week ago, the reading course on PL topology I'm going to follow started. The aim of the reading course is to understand the basics of PL topology and have a reasonable to good understanding of ...
1
vote
1answer
83 views

Einstein manifolds and topology

Given a Riemannian manifold $(M,g)$ with Ricci tensor $ R_{mn} = k g_{mn} $. Suppose the Ricci scalar you get is $$ R > 0 $$ What can you tell about the manifold $globally$ ? In particular, can ...
1
vote
1answer
28 views

Why is the sectional shape of a simply connected, oriented 4-manifold an isomorphism?

Let $M$ be a simply connected, compact and $\mathbb{Z}$-oriented 4-dimensional manifold. Let $\mu\in H_4(M;\mathbb{Z})$ be a fundamental class of $M$ (here is $H_4(M;\mathbb{Z})$ the singular homology ...
1
vote
1answer
83 views

Isogenies and dimensions

Let $f: \mathbb{C}^g/L\to\mathbb{C}^{g'}/L'$ be an isogeny of complex tori, i.e. a surjective Lie group morphism with finite kernel. Is it obvious that $g\ge g'$ ? It is easy to show that $f$ is ...
1
vote
1answer
78 views

Pullback of support of differential form

Let $f:X \to Y$ be a diffeomorphism between smooth complex compact manifolds. Let $\omega$ be a differential form on $Y$. It is true that the support of $f^*\omega$ is equal to $f^{-1}$ of the support ...