Questions tagged [manifolds]
A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.
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Uniqueness of compactification of an end of a manifold
Let $M$ be an $n$-dimensional manifold (smooth or topological). I call $\bar{M}$ a compactification of $M$ if it is an $n$-dimensional compact manifold with boundary $\partial \bar{M}$, an $(n-1)$-...
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G-spaces and manifolds
In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms:
The space is metric
The space is finitely compact, i.e., a ...
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Not all manifolds can be triangulated: In which dimensions?
I know that Ciprian Manolescu has settled the triangulation conjecture in the negative: Not all manifolds can be triangulated. I've only read secondary literature on this result, which did not detail ...
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When is a submanifold of $\mathbf R^n$ given by global equations?
Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...
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Is there a manifold with fundamental group $\mathbb{Q}$?
It is known that the fundamental group of a locally path connected, path connected compact metric space is finitely presented or uncountable. Furthermore the fundamental group of every manifold is ...
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3-fold of general type homeomorphic to rational 3-fold
Is there a smooth (complex projective) 3-fold of general type which is homeomorphic (in the complex topology) to a rational $3$-fold?
I am aware of such examples in complex dimension $2$, for ...
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When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?
I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for ...
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rational homotopy of a manifold
Given a finite dim rational homotopy type satisfying Poincaré duality,
what is the best reference to when it is the rational homotopy type of a fin dim manifold?
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Are there examples of non-orientable manifolds in nature?
Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence:
"The unorientable surfaces are never discussed ...
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Status of PL topology
I posted this question on math stackexchange but received no answers. Since I know there are more people knowledgeable in geometric and piecewise-linear (PL) topology here, I'm reposting the question. ...
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Is there a Whitney Embedding Theorem for non-smooth manifolds?
For smooth $n$-manifolds, we know that they can always be embedded in $\mathbb R^{2n}$ via a differentiable map. However, is there any corresponding theorem for the topological category? (i.e. Can ...
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Connected sum of topological manifolds
A definition of the connected sum of two $n$-manifolds $M$ and $M'$ begins by considering two $n$-balls $B$ in $M$, $B'$ in $M'$, and glueing the varieties $M\setminus \mathring B$ and $M'\setminus \...
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Classification of 1-dimensional manifolds (not second-countable)
It is easy to see that every connected $1$-dimensional second-countable manifold (that is, what is often called just a manifold) is either homeomorphic to $\mathbb{R}$ or to $S^1$. Now let's drop the ...
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Does every vector bundle allow a finite trivialization cover?
Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$.
(a) Is ...
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Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes
I search for a chain of clean references, which lead the fact of topological manifolds of dimension $n$ having the homotopy type of a CW of dimension $n$.
Milnor's On spaces having the homotopy type ...
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Algebraic varieties which are topological manifolds
Inspired by this thread, which concludes that a non-singular variety over the complex numbers is naturally a smooth manifold, does anyone know conditions that imply that the topological space ...
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Characterizing Hessians among symmetric bilinear tensors
I apologize in advance if this is somewhat elementary, but:
Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ ...
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Proving the existence of good covers
Usually one proves the existence of good covers in compact manifolds by Riemannian methods: we pick an arbitrary Riemannian metric, prove that geodesically convex neighborhoods exist, that they are ...
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Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex
Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$.
Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a CW-...
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Universal covers of non-prime 3-manifolds
Let $M$ be a closed, connected, oriented 3-manifold. If $M$ is prime, then we know what the universal cover of $M$ looks like: it is either $S^3, \mathbb{R}^3$ or $S^2 \times \mathbb{R}$ depending on ...
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Any 3-manifold can be realized as the boundary of a 4-manifold
We know
"Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
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What manifolds can have a (non-piecewise) linear structure?
By the definition I'm using, all manifolds are Hausdorff and second countable. For all non-negative integers $n$, I define $B_n$ to be $\bigl\{ \mathbf{v} \in \mathbf{R}^n : \lVert\mathbf{v}\rVert <...
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If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ be a triangulable manifold?
If $X_d$ is a non-triangulable manifold, can $X_d \times T^k$, $X_d \times I^k$, or $X_d \times \mathbb{R}^k$ always be a triangulable manifold?
Let $X_d$ be a $d$-manifold which is NOT a ...
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Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure
The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) ...
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Density of continuous functions to interior in set of all continuous functions
Let $M$ be an $m$-dimensional manifold and $N$ be an $n$-dimensional manifold with boundary. Suppose also that the topology on $N$ can be described by a metric. Thus, the set $C(M,N)$ can be endowed ...
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What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?
I know the following facts. (Don't assume I know much more than the following facts.)
The Atiyah-Singer index theorem generalizes both the Riemann-Roch theorem and the Gauss-Bonnet theorem.
The ...
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What is the difference between holonomy and monodromy?
What is the difference between holonomy and monodromy?
And what is the simplest example in which one is trivial and the other is not?
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Do rings of smooth functions differ from rings of continuous functions?
Let $M$, $N$ be connected nondiscrete compact smooth manifolds. Can the ring of continuous functions on $M$ be isomorphic to the ring of smooth functions on $N$?
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Are there only countably many compact topological manifolds?
Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be ...
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Good covers of manifolds
It is well-known and easy to prove (see for instance this post) that every smooth manifold admits a good cover, i.e., a locally finite cover by open balls such that all nonempty intersections of the ...
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A manifold is a homotopy type and _what_ extra structure?
Motivation: Surfaces
Closed oriented 2-manifolds (surfaces) are "classified by their homotopy type". By this we mean that two closed oriented surfaces are diffeomorphic iff they're homotopy ...
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Questions on J. F. Nash's answer about his errors in the proof of embedding theorem
In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked
Is it true, as rumours have it, that
you started to work on the embedding ...
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Are topological manifolds homotopy equivalent to smooth manifolds?
There exist topological manifolds which don't admit a smooth structure in dimensions > 3, but I haven't seen much discussion on homotopy type. It seems much more reasonable that we can find a smooth ...
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Non-regular Connected Hausdorff Banach Manifold
After reading this MO post, I am wondering:
Is every (connected) Hausdorff Banach manifold a regular space?
Though unjustified, page 53 of this paper nonchalantly states: "Note that a Hausdorff ...
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Is every rational realized as the Euler characteristic of some manifold or orbifold?
Let me first ask the question for two-dimensional compact, connected manifolds and orbifolds.
Then, if the answer is No, one can remove various conditions on the dimension,
and allow non-compact ...
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Are homeomorphic open subsets of $\mathbb{R}^n$ also diffeomorphic?
Let $U_1, U_2$ be open subsets of $\mathbb{R}^n$. Both are naturally differentiable submanifold, getting the differentiable structure from $\mathbb{R}^n$. Further, both are natural topological ...
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What is an example of an orbifold which is not a topological manifold?
In Thurston's book The Geometry and Topology of Three-Manifolds it is proven that the underlying space of a two-dimensional orbifold is always a topological surface.
Are there any easy examples of ...
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When is a finite cw-complex a compact topological manifold?
I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-...
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Is every real n-manifold isomorphic to a quotient of $\mathbb{R}^n$?
I'm curious about the following:
Is every real $n$-manifold isomorphic to a quotient of $\mathbb{R}^n$?
Thanks.
EDIT: As Tilman points out, the manifold should be connected. Also, yes, I'm thinking ...
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Is symmetric power of a manifold a manifold?
A Hausdorff, second-countable space $M$ is called a topological manifold if $M$ is locally Euclidean. Let $SP^n(M): = \left(M \times M \times \cdots \times M \right)/ \Sigma_m$, where product is done $...
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Homology spheres and fundamental group
I have a curiosity about homology spheres: I was wondering if they were uniquely characterized by their fundamental group. I.e. given two $n-$dimensional (integral) homology spheres with isomorphic ...
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Proofs of Rohlin's theorem (an oriented 4-manifold with zero signature bounds a 5-manifold)
A celebrated theorem of Rohlin states the following
An oriented closed 4-manifold $M^4$ bounds an oriented 5-manifold if and only if the signature of $M^4$ is zero.
Simple homological arguments ...
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If all balls around two points are isometric... -- manifold version
This question is a natural follow-up of this other question, asked earlier today by wspin.
Let's say that a metric space $(X,d)$ has two poles if:
there are two distinct points $x$, $y$ such that ...
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Does injectivity of $\pi_1(\partial U) \to \pi_1(M)$ imply injectivity of $\pi_1(U) \to \pi_1(M)$?
Let $M$ be a smooth compact manifold of dimension $n$, and let $U$ be a smooth compact manifold with boundary, of the same dimension $n$, embedded in $M$.
The embedding induces maps on $\pi_1$.
If $...
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homotopy type of embeddings versus diffeomorphisms
Previously, I asked a question on mathoverflow comparing smooth embeddings and diffeomorphisms, which received a very interesting and somewhat unexpected answer by Agol. I now ask a further question ...
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Tubular Neighborhood Theorem for $C^1$ Submanifold
Can anyone reference/disprove the theorem in the case where the embedded submanifold is merely $C^1$ instead of smooth? I have a compact $C^1$ embedded submanifold of $\mathbb{R}^n$ without boundary ...
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Obstruction of spin-c structure and the generalized Wu manifods
Bockstein homomorphim and obstruction of spin-c structure: Let $w_2$ be the Stiefel Whintney class of manifold $M$. Let the Bockstein homomorphim $\beta$ be the
$$
H^2(\mathbb{Z}_2,M) \to H^3(\mathbb{...
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Classification of 3-dimensional manifolds with boundary
It is well-known that every closed, connected and orientable 3-manifold $\mathcal{M}$ can uniquely be decomposed as
$$\mathcal{M}=P_{1}\#\dots\# P_{n}$$
where $P_{i}$ are prime manifolds, i.e. ...
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Are there examples of Einstein manifolds with unbounded curvature?
Are there any known examples of Einstein manifolds $(M, g)$ such that $$\sup_{x \in M} \|\text{Rm}(x) \| = \infty$$
I'm looking for these examples because they might provide a counter-example to a ...
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Closed manifolds with the fixed point property
The real projective plane ${\bf P}^2({\bf R})$ is the only closed surface with the fixed point property: all continuous maps from the plane to itself has a fixed point. The projective spaces ${\bf P}^{...