Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

4
votes
0answers
68 views

Lower bound on $\epsilon$-covers of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every point $...
-4
votes
0answers
49 views

Reachability of points on Manifolds Ask [on hold]

I was thinking with a friend that on a surface some points are more reachable than other.In the sense that their average distance to the other points is lower. e.g. suppose that we have a circle in a ...
5
votes
2answers
404 views

Classification of closed 3-manifolds with finite first homology group?

I am interested in a topological classification of connected closed 3-manifold $M$ that have finite homology group $H_1(M)$. Since $H_1(M)$ is the abelization of the fundamental group $\pi_1(M)$, ...
6
votes
1answer
250 views

Manifolds with negative dimension – Definition, References

Does the concept of differential manifold with negative dimension make sense, in differential geometry? If yes, how is it defined? Do you have any reference to recommend? My problem was born in ...
4
votes
1answer
197 views

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) ...
4
votes
1answer
125 views

The homological negligibility of certain subsets in compact manifolds

Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary). I need a reference to the following facts (which I believe are true at least in dimension $n=3$): Fact 1. For every ...
3
votes
1answer
144 views

Poincaré dual of the generators of $H^d(\mathbb{RP}^5,\mathbb{Z}_2)$

We know $H^d(\mathbb{RP}^5,\mathbb{Z}_2)=\mathbb{Z}_2$. So there are two classes of $\mathbb{Z}_2$ generators, trivial and nontrivial, for $d=0,1,2,3,4,5$. Wha are the Poincaré dual $(5-d)$-...
3
votes
0answers
42 views

Irreducible separators of compact manifolds

Definition. A closed subset $S$ of a topological space $X$ is called $\bullet$ a separator of $X$ if $X\setminus S$ is disconnected; $\bullet$ an irreducible separator if $S$ is a separator of $X$ ...
14
votes
0answers
216 views

Beyond smoothness-the clear picture about the notion of a differential form

In this paper N.Teleman constructs the signature operator on an arbitrary (closed, oriented) Lipschitz manifold with coefficients in a vector bundle $\xi$. In particular the notion of a differential $...
4
votes
0answers
113 views

Non-spin 5-manifold and $2^2$-Bockstein homomorphism

The $2^2$-Bockstein is $\beta_4$ is associated to $$0\to\mathbb{Z}/2\to\mathbb{Z}/{8}\to\mathbb{Z}/{4}\to 0,$$ (The $2^n$-Bockstein homomorphism $$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\...
2
votes
0answers
193 views

Quotient space, homogeneous space, and higher homotopy groups

Preparation and my input: For the quotient space $G/H$, knowing the homotopy groups of $G$ and $H$ one can determine homotopy groups from the long exact sequence $$ ... \to \pi_n(H) \to \pi_n(G) ...
1
vote
0answers
112 views

Completing the proof of that the set of points where $f(x) = 0$ is a $k$-manifold [closed]

[I have asked this question with the previous versions of my answer in math.SE; however, I did not get any comment / answer, so I thought I might asked this in here with the improved version of my ...
1
vote
1answer
144 views

Can a continuous map on a Hilbert manifold be approximated by a map which has infinitely many critical points?

It is well-known that a continuous map $f:M\to\mathbb{R}^n$ from a Hilbert manifold can be closely approximated by a smooth map $g:M\to\mathbb{R}^n$ which has no critical points. But, can such a ...
11
votes
1answer
428 views

Poincare duality spaces vs. manifolds via lifting maps, the obstruction theory and the role of simply connectedness

Suppose that we are given a topological space $X$: assume for simplicity that $X$ is compact we want to adress the following question: Is it true that one can find a manifold $M$ which is homotopy ...
12
votes
1answer
304 views

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{...
7
votes
1answer
221 views

Non-triangulable 4-manifold as a boundary of some 5 manifold

We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold. Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What ...
5
votes
2answers
421 views

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 ...
11
votes
1answer
361 views

Is there a PL, or topological, bordism hypothesis?

The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with ...
11
votes
3answers
587 views

Existence of non-null-homotopic map from $M^n$ to $S^{n-1}$

Let $M^n$ be compact, connected, oriented $n$-dimensional smooth manifold without boundary, the Hopf degree theorem states that the homotopy class of continuous maps from $M^n$ to $S^n$ is classified ...
1
vote
0answers
57 views

Does this “algebraic” method for the application of the constructive proof of the classification of closed & compact surfaces have any use? [closed]

Disclaimer: This question is cross-posted in here. I have never asked a question in mathoverflow before, so if the level of this question is not appropriate for this site, please just vote close it. ...
3
votes
0answers
34 views

Theoretical justification of time-series forecasting using Takens' embedding

This is a cross-posting where I couldn't get an answer. In the meantime I have tried to improve the original logic: As in Takens original paper about his embedding theorem, consider a compact $m$-...
8
votes
3answers
329 views

Is there a discrete lattice analogue of conformal transformations?

There is a simple discrete combinatorial analogue of manifolds and homeomorphisms: Replace manifolds by simplicial complexes and homeomorphisms by Pachner moves. Equivalence classes of manifolds under ...
7
votes
2answers
198 views

Are there invariants of cell complexes similar to the Euler characteristic?

The Euler characteristic is an invariant (under homeomorphism) of manifolds that can be computed from a cellulation by (weighted) counting of different kinds of objects, namely \begin{equation} \chi=\...
16
votes
1answer
338 views

Lowest Dimension for Counterexample in Topological Manifold Factorization

Bing gave a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is ...
10
votes
3answers
216 views

Do $\mathbb{HP}^2\#\overline{\mathbb{HP}^2}$ and $\mathbb{OP}^2\#\overline{\mathbb{OP}^2}$ arise as sphere bundles over spheres?

Recall that $\mathbb{RP}^2\#\mathbb{RP}^2$ is the Klein bottle and can be seen as a non-trivial $S^1$-bundle over $S^1$. In particular, it is the total space of the sphere bundle of $\gamma\oplus\...
0
votes
0answers
91 views

Why according to Takens' theorem (1981) we need 2m+1 observation functions (or time series) to reconstruct the attractor?

In the paper [1], page 369 we have Theorem 1 which says: Let $M$ be a compact manifold of dimension $m$. For pairs $(\phi,y),\;\phi: M\to M $ a smooth diffeomorphism and $y: M \to \mathbb{R}$ is a ...
2
votes
0answers
58 views

Transfer map between simplicial manifolds

Let $M^m$ and $N^n$ be two triangulated oriented and closed manifolds and $f:M\to N$ a simplicial map. For each $a\in H_p(N)$ we may consider its homological transfer $f_!a\in H_{m-n+p}(M)$. I want ...
3
votes
2answers
164 views

Signature of the manifold of the multiple fibrations over spheres

We can define the signature of a manifold in $4k$ dimensions. 1) If I understand correctly, the signature $\sigma$ of the manifold of the product space of spheres would always be zero: $$\sigma(S^...
6
votes
0answers
144 views

Differential topology on arbitrary fields

What do the differential topology theories on arbitrary fields have in common? Different differential topology theories There is "ordinary" differential topology on real manifolds, with its rich ...
1
vote
0answers
32 views

Criterion for bisemisimplicial sets to be a manifold

It is well known that a finite simplicial complex is a manifold of dimension $n$ iff the the link of each vertex is homeomorphic to $\mathbb{S}^{n-1}$. Are there any criteria for weaker structures? E....
3
votes
1answer
140 views

Arcwise-connectedness generalized to higher connectivity?

This is a crosspost from stackexchange. I'm not completely sure whether the question below is research-level, but I have not yet found an obvious answer, and what I have found thus far suggests that ...
1
vote
0answers
156 views

What intuitive concepts in pure math can be used to understand Big data? [closed]

I am not a mathematician but I need mathematicians' general knowledge and that is why I chose this community to ask my question from. As a student/researcher in Data Mining, with background in Pure ...
2
votes
0answers
91 views

Uniqueness of spheres in prime decomposition of a 3-manifold

Let $M$ be a closed connected orientable 3-manifold. Then Kneser tells us that there is a decomposition $M = P_1 \sharp \cdots \sharp P_k$ of $M$ into prime manifolds. Milnor tells us that if $M = ...
6
votes
1answer
404 views

What is “topology in dimension 3.5”?

I've noticed a couple of conference titles which reference something called "topology in dimension 3.5," such as this one and this one. This subject seems quite mysterious to me — it looks like ...
4
votes
0answers
102 views

Can non-chiral 3D TQFTs be extended to non-orientable manifolds whereas chiral ones cannot?

As far as I know, when talking about TQFT, one usually means TQFTs on oriented manifolds with boundary (cobordisms) It appears to me that the Turaev-Viro-Barrett-Westbury state-sum construction can ...
4
votes
3answers
167 views

Subset of $G_1(\mathbb{R}^n)$ having a line in common with every hyperplane of $G_{n-1}(\mathbb{R}^n)$

I am currently working on some problems related to Grassmann manifolds and eventually come to the following question. Let $S$ be a subset of $G_1(\mathbb{R}^n)$ such that any element of $G_{n-1}:=...
3
votes
0answers
76 views

“Signature Changing” Generalization of Lie Algebra?

I have in mind a mathematical structure I've never heard of before. Does anyone know what might be? It is a manifold with vector fields whose Lie brackets have structure coefficients that are ...
1
vote
0answers
129 views

Geometric quotient becomes a quotient manifold when passing to rational points

Let $k$ be a local field of characteristc zero. I'm interesting in understanding how morphisms of schemes of finite type over $k$ become morphisms of analytic manifolds on passing to rational points. ...
-3
votes
1answer
109 views

Randomizing on manifolds [closed]

Why we can't randomize on each manifold? If we have differential manifold, then using a map and parametrization we can randomize on it (for example on 2-dimensional torus). Where we have a problem in ...
7
votes
1answer
312 views

Can we cut and rotate a particular region of a hyperbolic 3-manifold to get another (non-homeomorphic) hyperbolic 3-manifold?

I'm trying to learn more about hyperbolic 3-manifolds, in particular the geometric implications of doing hyperbolic Dehn surgery to suitable knot complements. Following this paper by Christian ...
5
votes
1answer
266 views

Is the connected sum of a triangulable manifold with a non-triangulable manifold a non-triangulable manifold?

Let $M,N$ be topological manifolds such that $M$ does not admit a $PL$ structure and $N$ does. Is $M\#N$ still a triangulable manifold?
9
votes
1answer
380 views

Which topological manifolds do not correspond to strongly Hausdorff locales?

I'm toying with the idea of using locales as a way to define topological manifolds without beginning with points, largely for philosophical reasons. In this context I think I want to redefine a ...
5
votes
1answer
149 views

Continuously varying the singularities of a vector field

An arc field on a topological space $X$ is a continuous function $\Psi: X \rightarrow X^{[0,1]}$ such that for every $x \in X$, the path $\Psi(x): [0,1] \rightarrow X$ (1) starts at $x$, (2) is ...
2
votes
0answers
33 views

On the minimum distance along an orbit

Let $\Gamma$ be a nontrivial group of isometries of $\mathbb{S}^n$, $n \geq 2$, acting properly discontinuously. For $p \in \mathbb{S}^n$, define $$r(p) = \min_{g \in \Gamma \setminus\{e\} } d(p, g(p)...
10
votes
2answers
515 views

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 ...
7
votes
0answers
167 views

Atiyah duality with coefficients and boundary

Looking at Atiyah's paper "Thom complexes", I find two statements of the Atiyah duality theorem: Proposition 3.2 Let $X$ be a compact smooth manifold with boundary $Y$ and tangent bundle $\tau$. Then ...
13
votes
1answer
436 views

A question on connected sum of compact manifolds

Let $M$ be a compact orientable manifold which is homeomorphic to its connected sum with itself $M\# M$. Must $M$ be homeomorphic to a sphere? I will explain why I am interested (at the risk of being ...
2
votes
0answers
65 views

When does the gradient of a solution to elliptic equation vanish on the boundary?

This question is motivated by an inverse coefficient problem, for which it is useful to find solutions to a particular PDE so that the gradient of the solution does not vanish at all, or at least too ...
2
votes
0answers
55 views

Tangent space of a linear section

Let $X\subset\mathbb{C}^n$ be an embedded smooth manifold, $Y = X\cap H$ a smooth hyperplane section of $X$, $p\in Y\subset X$ a point, and $T_pX, T_pY$ the embedded tangent spaces at $p$ of $X$ and $...
6
votes
1answer
239 views

Fundamental groups of non-orientable closed four-manifolds

The fundamental group of a closed orientable manifold is finitely presented, and every finitely presented group arises as the fundamental group of a closed orientable four-manifold; see this question. ...