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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143 votes
20 answers
23k views

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 ...
107 votes
8 answers
15k views

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 ...
Qiaochu Yuan's user avatar
95 votes
4 answers
10k views

Which manifolds are homeomorphic to simplicial complexes?

This question is only motivated by curiosity; I don't know a lot about manifold topology. Suppose $M$ is a compact topological manifold of dimension $n$. I'll assume $n$ is large, say $n\geq 4$. The ...
Charles Rezk's user avatar
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59 votes
7 answers
7k views

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. ...
55 votes
8 answers
9k views

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 ...
Jake's user avatar
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51 votes
3 answers
11k views

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?
James Propp's user avatar
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46 votes
3 answers
8k views

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 \...
ACL's user avatar
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43 votes
4 answers
3k views

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 $N$?
Arshak Aivazian's user avatar
41 votes
4 answers
4k views

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 ...
Oblomov's user avatar
  • 2,501
40 votes
1 answer
2k views

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 ...
Dominik's user avatar
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39 votes
1 answer
6k views

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 ...
Joseph O'Rourke's user avatar
38 votes
4 answers
4k views

What manifolds are bounded by RP^odd?

Real projective spaces $\mathbb{R}P^n$ have $\mathbb{Z}/2$ cohomology rings $\mathbb{Z}/2[x]/(x^{n+1})$ and total Stiefel-Whitney class $(1+x)^{n+1}$ which is $1$ when $n$ is odd, so it follows that ...
Pierre Weil's user avatar
35 votes
2 answers
4k views

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 ...
Misha's user avatar
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34 votes
7 answers
15k views

geometric interpretation of Lie bracket

On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written: We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to which the integral ...
Sepideh Bakhoda's user avatar
32 votes
1 answer
1k views

"Affine communication" for topological manifolds

There is a situation that comes up regularly in algebraic topology when giving proofs of facts about manifolds, like Poincare duality and the like. The typical sequence goes like this: Prove ...
Tyler Lawson's user avatar
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31 votes
1 answer
4k views

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 ...
123...'s user avatar
  • 663
31 votes
3 answers
2k views

A Pachner complex for triangulated manifolds

A theorem of Pachner's states that if two triangulated PL-manifolds are PL-homeomorphic, the two triangulations are related via a finite sequence of moves, nowadays called "Pachner moves". A ...
Ryan Budney's user avatar
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30 votes
2 answers
2k views

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 ...
Manuel Bärenz's user avatar
29 votes
15 answers
5k views

Important results that use infinite-dimensional manifolds?

Are Banach manifolds (or other types of infinite-dimensional manifolds) just curiosities, or have they been utilized to prove some interesting/important results? Where do they turn up? Important ...
29 votes
3 answers
3k views

A book on locally ringed spaces?

Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a ...
28 votes
3 answers
2k views

A "meta-mathematical principle" of MacPherson

In an appendix to his notes on intersection homology and perverse sheaves, MacPherson writes Why do we want to consider only spaces $V$ that admit a decomposition into manifolds? The intuitive ...
Faisal's user avatar
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27 votes
2 answers
3k views

Euler Characteristic of a manifold with non-vanishing vector field,

A friend of mine recently asked me if I knew any simple, conceptual argument (even one that is perhaps only heuristic) to show that if a triangulated manifold has a non-vanishing vector field, then ...
Dick Palais's user avatar
  • 15.2k
27 votes
2 answers
731 views

Is there a flat manifold with trivial first homology?

Is there a closed flat manifold whose fundamental group has trivial abelianization? The famous Hantzsche–Wendt flat manifold has fundamental group with finite abelianization.
Igor Belegradek's user avatar
26 votes
4 answers
785 views

Ring of closed manifolds modulo fiber bundles

Let $R$ be the ring which is generated by homeomorphism classes $[M]$ of compact closed manifolds (of arbitrary dimension) subject to the relations that $$[F]\cdot [B] = [E]$$ if there exists a fibre ...
Andreas Thom's user avatar
  • 25.3k
26 votes
1 answer
908 views

Closed manifold with non-vanishing homotopy groups and vanishing homology groups

Is there a closed connected $n$-dimensional topological manifold $M$ ($n\geq 2$) such that $\pi_i(M)\neq 0$ for all $i>0$ and $H_i(M, \mathbb{Z})=0$ for $i\neq 0$, $n$? The manifold $S^1\times S^2$ ...
user avatar
26 votes
1 answer
4k views

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 ...
Martin Brandenburg's user avatar
26 votes
2 answers
2k views

Are there geometrically formal manifolds, which are not rationally elliptic?

Formality of a space is meant in the sense of Sullivan, i.e. a space $X$ is called formal, if its commutative differential graded algebra of piecewise linear differential forms $(A_{PL}(X),d)$ is ...
archipelago's user avatar
  • 2,964
25 votes
2 answers
4k views

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 ...
trequartista's user avatar
25 votes
1 answer
1k views

On a curious map from the complex projective plane into $S^5$

I have heavily edited the post (including the title), based on a comment by @GregoryArone that my map $f$ is not injective. In an earlier version of this post, I had thought to have constructed a ...
Malkoun's user avatar
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25 votes
1 answer
1k views

Smooth manifolds as idempotent splitting completion

The nlab has a particularly interesting thing to say about the category of smooth manifolds: it is the idempotent-splitting completion of the category of open sets of Euclidean spaces and smooth maps. ...
Arrow's user avatar
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24 votes
2 answers
3k views

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 ...
Jake's user avatar
  • 795
24 votes
3 answers
2k views

Gauss-Bonnet Theorem for Graphs?

One can define the Euler characteristic χ for a graph as the number of vertices minus the number of edges. Thus an $n$-cycle has $\chi = 0$ and $K_4$ has $\chi=-2$. Is there an analog for the ...
Joseph O'Rourke's user avatar
24 votes
1 answer
1k views

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 ...
Benjamin Dickman's user avatar
24 votes
3 answers
2k views

The third axiom in the definition of (infinite-dimensional) vector bundles: why?

Serge Lang's Differential and Riemannian Manifolds is a no doubt the best available reference for the theory of not-necessarily-finite-dimensional differential manifolds, but unfortunately it suffers ...
slow student's user avatar
23 votes
2 answers
1k views

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 ...
Joseph O'Rourke's user avatar
23 votes
2 answers
2k views

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 $(n-1)$-...
Igor Khavkine's user avatar
23 votes
3 answers
2k views

Does a *topological* manifold have an exhaustion by compact submanifolds with boundary?

If $M$ is a connected smooth manifold, then it is easy to show that there is a sequence of connected compact smooth submanifolds with boundary $M_1\subseteq M_2\subseteq\cdots$ such that $M=\bigcup_{i=...
John Pardon's user avatar
  • 18.3k
23 votes
2 answers
2k views

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 ...
Mark Ullmann's user avatar
23 votes
1 answer
1k views

Mapping class groups in high dimension

$\DeclareMathOperator\MCG{MCG}\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\Homeo{Homeo}$Let $M$ be a $1$-connected, closed, smooth manifold with $\dim(M)>4$ and let us set $\MCG(M)=\pi_0(\...
David C's user avatar
  • 9,792
23 votes
0 answers
652 views

Do most manifolds have symmetries? or not?

Let us say that a (closed, connected) manifold has a symmetry if it admits a non-trivial action by a finite group. Note that I am not asking the action to be free. So for example rotating the 2-sphere ...
Chris Schommer-Pries's user avatar
22 votes
6 answers
3k views

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 ...
Fiktor's user avatar
  • 1,264
22 votes
2 answers
2k views

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 ...
user88649's user avatar
  • 271
21 votes
1 answer
724 views

Is $\mathbb{CP}^3$ minus two points the universal cover of a compact manifold?

After reading some recent questions on mathoverflow about universal coverings, I am curious about the following: Is it possible to construct a closed $6$-manifold $M$, with universal cover ...
Nick L's user avatar
  • 6,923
21 votes
2 answers
4k views

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 ...
Tom's user avatar
  • 489
20 votes
1 answer
1k views

isotopy inverse embeddings vs. diffeomorphisms

I would like to find an example, if one exists, of manifolds $M$ and $N$ with embeddings $f:M\to N$ and $g:N\to M$ such that $f\circ g$ and $g\circ f$ are both isotopic (i.e. homotopic through ...
Ricardo Andrade's user avatar
20 votes
2 answers
2k views

If $X$ is a simplicial complex, is there a characterization of the links of its vertices that is equivalent to the statement "$|X|$ is a manifold"?

We have a characterization when we want $|X|$ to be a PL-manifold, in particular that the links of all the vertices are themselves (PL) spheres. If we are in the category of PL- spaces then this is a ...
Spice the Bird's user avatar
20 votes
1 answer
564 views

Manifolds as Cauchy completed objects

The category of smooth manifolds (SmoothMfld) can be thought of the Cauchy completion of the category $U$ of open subsets of Euclidean spaces (with smooth maps) [1]. This fact is shocking to me as it ...
Student's user avatar
  • 5,008
20 votes
2 answers
859 views

Distinct manifolds with the same configuration spaces?

For a space $X$, let $C_k X$ denote the space of configurations of $k$ distinct unordered points in $X$. What is an example of a pair of smooth manifolds $M$ and $N$ that are not homeomorphic but ...
cdouglas's user avatar
  • 3,083
18 votes
5 answers
2k views

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 ...
Eivind Dahl's user avatar
18 votes
4 answers
3k views

(Very) High dimensional manifolds

Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...

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