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
55 views

Induced new structures on Poincare dual manifolds

"R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$...
6
votes
0answers
130 views

GSO (Gliozzi-Scherk-Olive) projection and its Mathematics?

GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the ...
3
votes
0answers
38 views

Relation between the orientation sheaves of the interior and the boundary of a topological manifold

Let $(M, \partial M)$ be an $n$-dimensional topological manifold with boundary. Let $\mathcal{O}_{M \setminus \partial M}$ and $\mathcal{O}_{\partial M}$ denote the orientation sheaves of $M \setminus ...
5
votes
1answer
125 views

Relating bordism groups of different dimensions

Let $M_d$ be a $d$-manifold generator of a subgroup of bordism group $$ \Omega_d^{G}, $$ or further generalization $$ \Omega_d^{G}(K(\mathcal{G},n+1)), $$ which $G$ is the given structure ...
8
votes
3answers
889 views

Is there a closed non-smoothable 4-manifold with zero Euler characteristic?

I will just repeat the title: Is there a closed non-smoothable 4-manifold with zero Euler characteristic? I am guessing yes simply based on other existence theorems I have seen for 4-manifolds.
1
vote
0answers
43 views

Upper bounds on $\epsilon$-covers of arbitrary compact manifolds

Let $M \subset \mathbb{R}^d$ be a compact $k$-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 ...
4
votes
0answers
87 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 $...
5
votes
2answers
475 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
320 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
239 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
131 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
159 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
43 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 $X$ ...
15
votes
0answers
276 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
124 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
206 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
115 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
148 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
441 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
332 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{...
8
votes
1answer
233 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 ...
6
votes
2answers
440 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
365 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
607 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
35 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
387 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
206 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
341 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
224 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
102 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
169 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
161 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
33 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
145 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
157 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
93 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
409 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
103 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
77 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
133 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
315 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
271 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
150 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
558 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 ...