All Questions
58 questions
3
votes
0
answers
119
views
Signature vs commensurability
If a closed oriented $4n$-dimensional manifold $M$ has an orientation-reversing homeomorphism, then the signature $\sigma(M)$ of the intersection form vanishes. More generally, $\sigma(M) = 0$ if $M$ ...
- 41
1
vote
0
answers
57
views
extendability of fibre bundles on manifolds with same dimensions
Let $M$ be an $m$-manifold. Let $M'\subseteq M$, where
$M'$ is also an $m$-manifold.
Let $N$ be an $n$-manifold. Let $N'\subseteq N$, where
$N'$ is also an $n$-manifold.
Suppose there is fibre ...
- 123
3
votes
3
answers
537
views
Fundamental group of a generalized connected sum
Let $M$ and $N$ be two $n$ dimensional connected closed manifolds with $n \ge 3$, and let $S$ be a $(n-1)$ dimensional closed submanifold common to $M$ and $N$. Consider the connected sum of $M$ and $...
- 311
0
votes
0
answers
120
views
Topological transversality by dimension
We know that to achieve transverality in the topological category, for example to make a continuous map into a manifold transverse to a topological submanifold, we need the existence of micro normal ...
- 803
8
votes
1
answer
484
views
Why does the tangent classifier classify the tangent (micro)bundle?
Let $\mathcal{M}\mathrm{fld}_n$ denote the $\infty$-category of topological manifolds (without boundary) and embeddings; more precisely, it is the homotopy coherent nerve of the simplicial category ...
- 2,292
8
votes
1
answer
281
views
Non-compact three-manifolds with the same proper homotopy type are homeomorphic?
I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not):
Let $M, M'$ be two non-compact connected $3$-manifolds with the ...
- 1,097
5
votes
0
answers
164
views
The fundamental group of the complement of badly embedded open $n$-ball in $\Bbb R^n$
Let $\mathcal D^n$ be an open subset of $\Bbb R^n$ such that $\mathcal
D^n$ is homeomorphic to $\{x\in \Bbb R^n:|x|<1\}$. Suppose $\Bbb R^n\setminus \mathcal D^n$ is path-connected. How bad can
$\...
- 1,097
6
votes
1
answer
304
views
Stable torus that is not a torus [duplicate]
Let $M$ be a closed manifold such that $M\times \mathbb{S}^1$ is a torus.
Is it true that $M$ is homeomorphic to a torus?
- 45k
13
votes
0
answers
223
views
Examples of manifolds with first nontrivial SW-class in degree 16 or bigger
As a module over the Steenrod algebra, $H^{\ast}(BO;\mathbb F_2) = \mathbb F_2[w_1, w_2, w_3, \dots]$ is generated by $w_{2^t}, t \geq 0$. Thus, the first nontrivial SW-class of any vector bundle $\xi ...
- 11.9k
8
votes
0
answers
290
views
When does the tangent microbundle of a closed orientable topological $4k$-manifold have a trivial rank 2 subbundle?
$\DeclareMathOperator{\Top}{Top}
\DeclareMathOperator{\co}{H}$Let $M$ be a closed orientable connected topological manifold of dimension $4k$ with $k > 1$. It is known (David Frank, On the index of ...
- 1,726
8
votes
1
answer
399
views
Universal cover with one end
Suppose that $M$ is a non-compact manifold of finite topological type with one end which is the universal cover of some closed manifold $N$.
Is $M $ necessarily homeomorphic to the total space of some ...
- 6,995
21
votes
1
answer
754
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 ...
- 6,995
5
votes
1
answer
297
views
Set of proper homotopy classes of arcs in a manifold
Let $M^n$ be an $n$-manifold with nonempty boundary and let $\partial_0 M$ be a specific connected component of $\partial M$. I am interested in the set of continuous maps $f : [0,1] \to M$ such that $...
- 5,349
8
votes
3
answers
548
views
Contractible set in a manifold
Let $M$ be an $n$-dimensional topological closed manifold. Suppose $K$ is a compact subset of $M$ which is contractible in the sense that there exists a continuous map $F:K \times [0,1] \to M$ with $F(...
- 891
3
votes
2
answers
191
views
Embedded submanifold in a cylinder
Let $M^n$ be an $n$-dimensional topological closed manifold. Suppose there exists an embedding $i:M \to M \times [0,1]$ such that $i(M)$ is contained in the interior of $M \times [0,1]$ and separates $...
- 891
17
votes
2
answers
1k
views
Suspension of a topological space
Let $X$ be a topological space such that its suspension is a topological manifold. Can we prove that $X$ itself is a topological manifold?
- 2,535
8
votes
2
answers
260
views
Degree one self-map of $\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$ not homotopic to any self-homotopy equivalence
Consider the surface $\Sigma=\Bbb R^2\big\backslash \big\{(n,0):n\in \Bbb Z\big\}$. Does there exist a proper
map $f\colon \Sigma\to \Sigma$ of degree $1$ and not homotopic to
any self-homotopy ...
- 265
6
votes
0
answers
209
views
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 ...
- 10.5k
25
votes
1
answer
2k
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 ...
- 5,215
6
votes
1
answer
506
views
Map which is null-homotopic on compacts
This is the missing ingredient towards answering my previous question.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). ...
- 5,529
9
votes
2
answers
755
views
Is limit of null-homotopic maps null-homotopic?
The question is motivated by my failed comment to this one.
Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds).
Let $\...
- 5,529
8
votes
1
answer
1k
views
Cobordism Theory of Topological Manifolds
Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds.
Cobordism Theory for DIFF/Differentiable/smooth manifolds
However, there are ...
- 10.5k
4
votes
0
answers
310
views
Topological approach to create a space between clouds
I have a dataset associated with labels. According to https://arxiv.org/pdf/1802.03426.pdf --> UMAP (Uniform Manifold Approximation and Projection) which is a novel manifold learning technique for ...
- 51
5
votes
0
answers
109
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 $\text{Pin}^-$...
- 10.5k
5
votes
1
answer
192
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 ...
- 10.5k
8
votes
2
answers
2k
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 $\pi_1(M)$, ...
- 41.9k
5
votes
1
answer
503
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) ...
- 10.5k
3
votes
1
answer
266
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 $(5-d)$-...
- 10.5k
4
votes
0
answers
241
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
$$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\...
- 10.5k
3
votes
0
answers
888
views
Quotient space, homogeneous space, and higher homotopy groups
Preparation and my input:
For the quotient space $G/H$, knowing the homotopy
groups of $G$ and $H$ one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(H) \to \pi_n(G) ...
- 10.5k
5
votes
1
answer
194
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 ...
- 1,726
41
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 ...
- 151k
6
votes
0
answers
395
views
Do all topological manifolds admit locally flat embeddings into R^n?
In his 1969 paper “Locally flat imbeddings of
topological manifolds” Lees proved that a closed oriented
second countable topological manifold admits a locally flat embedding into some R^n.
Does the ...
- 37.8k
10
votes
1
answer
557
views
Elements of infinite order in the topological mapping class group
Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}_0(...
- 18.7k
4
votes
1
answer
138
views
Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?
Suppose it is given an orientation preserving homotopy equivalence $h:N→M$ between closed oriented connected manifolds. Let $X,Y\subset M$ be diffeomorphic submanifolds, and assume $h$ to be ...
- 362
24
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(\...
- 9,870
8
votes
0
answers
222
views
Geometric argument for "easy" part of Jordan-Brouwer separation theorem without local flatness
Let $M^n \subset \mathbb{R}^{n+1}$ be an $n$-dimensional compact connected topological submanifold. The Jordan-Brouwer separation theorem says that $\mathbb{R}^{n+1} \setminus M^n$ contains two ...
- 321
6
votes
1
answer
482
views
Homeomorphism of closed manifold
Suppose that we have two closed n-manifold $M$ and $N$ such that
the topological group of homeomorphisms $Homeo(M)$ is homotopy equivalent to $Homeo(N)$ (maybe as topological groups if needed), can ...
- 277
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 ...
- 489
2
votes
0
answers
239
views
Are there analogs of smooth partitions of unity and good open covers for PL-manifolds?
Smooth partitions of unity and differentiable good open covers are important technical tools in the realm of smooth manifolds.
Are there analogs of these tools for piecewise linear manifolds?
A PL ...
- 37.8k
4
votes
1
answer
609
views
Manifolds such that every homeomorphism of a submanifold to itself extends to the full manifold
Let manifold $S$ (connected, without boundary) have next property: for every submanifold $D \subset S$ (connected, compact, without boundary), every homeomorphism $f:D \to D$ extends to a ...
- 1,307
5
votes
1
answer
1k
views
On the fundamental group of closed 3-manifolds
I know that every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). In dimension 2 there are strong restriction on ...
- 683
4
votes
0
answers
269
views
Action of the mapping class group on separating spheres in a connected sum of aspherical 3-manifolds
Let $M$ be a connected sum of $g$ closed aspherical 3-manifolds $M_1, \ldots, M_g$. [Update: I also assume that all the $M_i$-s are diffeomorphic, i.e. $M$ is a connected sum of copies of the same ...
- 3,203
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 ...
- 2,974
2
votes
0
answers
1k
views
What kinds of manifolds admit non-vanishing vector fields defining convergent congruences?
One of the corollaries of the Poincaré–Hopf index theorem is that a closed, connected manifold $M$ admits non-vanishing vector fields iff its Euler characteristic is zero; i.e. $\chi(M) = 0$.
I am ...
1
vote
0
answers
99
views
PL or projective PL map on the links of a PL manifold
Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
- 1,373
20
votes
2
answers
870
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 ...
- 3,103
28
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=...
- 18.7k
7
votes
1
answer
953
views
Cancellation law for $M^n\times \mathbb R= N^n\times \mathbb R$.
Assume $M^n$ and $N^n$ are null bordant, i.e. each can be realized as boundary of an $n+1$ dimensional manifold. Suppose $M^n \times \mathbb R$ is homeomorphic to $N^n\times \mathbb R$. Is there any ...
- 2,623
17
votes
2
answers
2k
views
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 ...
- 6,210