All Questions
97 questions
4
votes
0
answers
158
views
Cell structure on the function space $\operatorname{Hom}(X,Y)$
By the Theorem of Milnor in his paper "On spaces having the homotopy type of a CW-complex", the function space $\operatorname{Hom}(X,Y)$ (with the compact-open topology) is homotopy ...
- 150
6
votes
1
answer
206
views
A stable splitting of linear surjections
Some computations I've been doing in Weiss calculus predict the following stable splitting of the space of linear surjections: $\Sigma^\infty_+ \mathrm{Sur}(\mathbb{R}^n,\mathbb{R}^{m_1+m_2})$
as the ...
- 5,839
2
votes
0
answers
414
views
$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]
If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of
$$
\left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
- 705
8
votes
0
answers
151
views
The James and Morse filtrations of homotopy groups
Denote by $JX$ the James construction on a path connected, well-pointed space $X$. This space is filtered by subspaces $X=J_1X\subseteq\dots\subseteq J_nX\subseteq\dots\subseteq JX$ and is the domain ...
- 5,596
9
votes
1
answer
322
views
Torsion and nilpotence of framed manifolds under the Pontryagin-Thom map
The framed Pontryagin Thom construction produces a graded ring isomorphism from the framed cobordism ring $\Omega^{fr}_*$ to the stable ring of homotopy groups of spheres $\pi_*(\mathbb{S})$. I have a ...
6
votes
1
answer
414
views
Do $h$-cobordism groups arise from a 'Thom-like' spectrum?
Not thinking about $h$-cobordism, one usually defines a cobordism between manifolds, realizes it is an equivalence relation, chooses an appropriate class of structured manifolds (framed, unoriented, ...
- 609
2
votes
1
answer
127
views
Preservation of fiberwise normal bundles under fiberwise homotopy equivalences
I am looking to replicate, in the fiberwise setting, the result of Spivak/Wall that the fiber homotopy type of the normal sphere bundle of a manifold is preserved under homotopy equivalences.
A ...
- 5,839
2
votes
0
answers
118
views
Configurations of points in a spectrum
I am wondering if the following construction has appeared in the literature:
Let $X$ be a pointed space. Define the singular configuration space of $X$, $F^*(X,k)$ ,to be $\{(x_1,\dots,x_k)| x_i=x_j \...
- 5,839
3
votes
0
answers
113
views
Extending good covers of $\partial M$ to $M$
Suppose $M$ is an $n$-dimensional manifold with boundary with a free action of a finite group $G$. Suppose one has an equivariant collar $c: \partial M \times [0,1) \rightarrow M$. An open cover is ...
- 5,839
6
votes
0
answers
162
views
Uniqueness of normal microbundle of a smooth embedding
Suppose $M$ is a topological manifold and $\iota: N\hookrightarrow M$ be a submanifold. A normal microbundle of $N$ consists of an open neighborhood $U$ of $N$ and a retraction $\pi: U \to N$ such ...
- 803
9
votes
1
answer
588
views
Oriented bordism in higher dimensions (e.g. $12 \leq d \leq 28$)
The classification of oriented compact smooth manifolds up to oriented cobordism is one
of the landmarks of 20th century topology. The techniques used there form the part of the foundations of ...
- 10.5k
1
vote
0
answers
97
views
Homotopy type of complement to a union of linear subspaces
Im not sure if this question is appropriate for MO, but I'm looking for a hint about some questions about homotopy type of complement to a union of linear subspaces in vector space $\mathbb{R}^n, \...
- 11
5
votes
1
answer
372
views
$\pi_{2n-1}(\operatorname{SO}(2n))$ element represents the tangent bundle $TS^{2n}$, not torsion and indivisible for $n>1$?
Question: Is the element $\alpha$ in $\pi_{2n-1}(\operatorname{SO}(2n))$ representing the tangent bundle $TS^{2n}$ of the sphere $S^{2n}$ indivisible and not torsion?
My understanding so far —
An $\...
- 10.5k
3
votes
1
answer
115
views
Hadamard-like product on orientable surfaces
Denote by $C$ the category of connected closed orientable surfaces.
Is there a functor $F:C\times C\to C$ such that $b_1(F(S\times S'))=b_1(S)b_1(S')$?
- 41
12
votes
1
answer
489
views
Homological stability and Waldhausen A-theory
$\DeclareMathOperator{\Diff}{Diff}$
From the work of Galatius - Randall-Williams and Berglund - Madsen we have homological stability (with respect to g) of $B\Diff_\partial (W_{g,1})$ and rational ...
- 5,839
4
votes
0
answers
191
views
Isotopy classes of small codimension embeddings
Let $M^n$ be a smooth closed, oriented $n$-manifold.
Let $S_0,S_1\subset M^n$ two connected, compact and (positively) oriented submanifolds of $M$ of codimension $k$ diffeomorphic to $S$.
Suppose $k=...
- 1,040
3
votes
1
answer
124
views
Homotoping diffeomorphism to a $J$-holomorphic one
Let $M$ be a closed simply-connected smooth manifold. Assume $M$ admits at least one almost complex structure.
Is any diffeomorphism $M\to M$ homotopic as a continuous map to a $J$-holomorphic ...
user164740
18
votes
0
answers
496
views
Orientation-reversing homotopy equivalence
If there is an orientation-reversing homotopy equivalence on a closed simply-connected orientable manifold is there an orientation-reversing homeomorphism?
It is not true, for instance, that if there ...
user164740
3
votes
0
answers
98
views
Non-diffeomorphic surface bundles over homeomorphic 4-manifolds
For a smooth manifold $M$ an $M$-surface is the total space of a smooth surface bundle over $M$.
Let $M_1$ and $M_2$ be two homeomorphic closed simply-connected smooth 4-manifolds. Can there be an $...
user164740
11
votes
1
answer
379
views
Smooth structure on direct product
Let $M$ be the $E_8$ manifold. Is there a closed manifold $N$ such that $M\times N$ is smoothable? What is the smallest possible dimension of $N$?
user164740
9
votes
0
answers
333
views
Homotopical characterization of CW complexes
Let $X$ be a compact metrizable topological space of covering dimension $n\leq 3$.
Is it possible to give a necessary and sufficient condition for $X$ to be a CW complex in terms of the homotopy types ...
- 117
9
votes
0
answers
200
views
Homotopical characterization of manifolds
Let $X$ be a compact metrizable topological space of covering dimension $4$.
Assume that for any point $x\in X$ any neighbourhood of $x$ contains a contractible open neighbourhood $U$ such that $U\...
- 117
1
vote
0
answers
152
views
Complement of contractible locally Euclidean subspace
Let $X$ be a connected closed topological manifold. Let $S\subset X$ be a contractible locally Euclidean subspace. Is $X\setminus S$ connected?
- 11
1
vote
0
answers
154
views
Homotopy groups of ball complement
Let $X$ be a connected closed topological manifold. Let $n$ be an integer such that $\pi_i(X)=\{0\}$ for $1\leq i \leq n$.
Let $f:B^m\to X$ be a topological embedding, where $B^m$ is the $m$-...
- 19
1
vote
0
answers
137
views
Covers of a 4-manifold pull back a cohomology class to any algebraic multiple
Fix an algebraic integer $x\neq 0$. Is there a closed smooth 4-manifold $M$ with a class $\rho\in H^{1}_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^*\rho=x\rho$?
Is ...
user164740
4
votes
1
answer
222
views
Smooth covers pulling back a cohomology class to any algebraic multiple
Fix an algebraic integer $x\neq 0$. Does there exist a closed smooth manifold $M$ with a class $\rho\in H^{1}_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^*\rho=x\rho$?
user164740
9
votes
1
answer
349
views
Smooth complex projective surface as the total space of a Serre fibration
Let $M$ be the underlying topological manifold of a smooth complex projective surface. Assume $\pi_1(M)=\{0\}$ and $\pi_2(M)\neq \mathbb{Z}^2$.
Is there a Serre fibration $M\to B$ where $B$ is a CW ...
user164740
8
votes
0
answers
217
views
Hopf invariants of elements from spherical fibrations
Let $G_n$ be the space of homotopy-equivalences of $S^{n-1}$. Evaluation produces a map $G_{n} \to S^{n-1}$. For $n = 2m+1$, I would like to understand the induced map on $\pi_{4m-1}$. More precisely, ...
- 11.9k
13
votes
0
answers
330
views
One periodic cohomology theories?
Is there a classification of one periodic cohomology theories? In other words, spaces $X$ such that $\Omega X \simeq X$? For example, $X=*$ and $X= K(G,0) \times K(G,1) \times \dots$ are examples. ...
- 5,839
7
votes
1
answer
448
views
Computing $\pi_2$ of the complement of a 2-knot or spaces with aspherical splittings?
As far as I know, it is an open question if the complement of a ribbon disk $D^2 \subset B^4$ is aspherical. In reading "Some remarks on a problem of J.H.C Whitehead" by Howie, it is noted that there ...
- 5,349
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
19
votes
1
answer
790
views
Which cohomology classes are detected by tori?
Given a space $X$, I am looking for a characterization of classes $\alpha \in H^n(X;\bf Q)$ such that there is a map $f\colon T^n \to X$ so that $f^{\ast} \alpha$ pairs non-trivially against the ...
- 11.9k
12
votes
0
answers
408
views
The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from that of topological manifolds?
Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the ...
- 7,789
1
vote
0
answers
73
views
Cyclic homotopies of quotients of $S^3$
We are given a free action of an abelian finite group on $S^3$. Let $L$ denote the quotient space and let an element $\alpha \in \pi_1 L =G$ be given. Does there exist a cyclic homotopy $h_t:L \to L$ ...
- 268
18
votes
2
answers
1k
views
What is the generator of $\pi_9(S^2)$?
This is more or less the same question as
[ What is the generator of $\pi_9(S^3)$? ], except what I would like to know is if it is possible to describe this map in a way
not only topologists can ...
- 6,891
4
votes
1
answer
198
views
Space of non-vanshing sections path-connected?
Let $M$ be a path connected smooth manifold and $E$ be a vectorbundle over $M$ of rank at least two. My question is: Under which conditions is the space of global non-vanishing sections path connected?...
- 271
6
votes
0
answers
634
views
Quotient space, a fundamental group, and higher homotopy groups 2
Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...
- 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
0
answers
208
views
Asphericity of hypersurface complement in ${\mathbb C}^n$
How does one check that the following space is aspherical?
$X_n=\{(x_1,x_2,\ldots , x_n)\in {(\mathbb C^*)}^n\ |\ x_i\neq x_j\ and\ x_ix_j\neq 1\ for\ i\neq j\}$.
One way I can think of is to give ...
- 585
5
votes
0
answers
75
views
Bounding the dimension of the euclidean space in which any $n$-manifold embeds "$k$-uniquely" in it
(The question will be interesting for topological/Pl as well but in order to not be too vague I will restrict the meaning of manifold to smooth manifold without boundary).
I'm interested in the ...
- 7,789
3
votes
1
answer
293
views
Are there compact flat fiber bundles with "truly" infinite structure group?
Let $p: E \rightarrow B$ be a flat fiber bundle with fiber $F$ where $E$, $B$, $F$ are compact, smooth manifolds.
I am looking for a counterexample for the following statement:
$E \cong \widetilde{B}...
- 404
5
votes
0
answers
339
views
What is the local structure of a fibration?
It's sometimes said that a fibration is a fiber bundle which is not locally trivial. I'd like to make this precise, by identifying the "local models" on which fibrations are modeled.
Here I'd like ...
- 63.9k
3
votes
0
answers
118
views
Weak contractibility of some infinite dimensional metric spaces
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$, ...
10
votes
4
answers
2k
views
Complements of Simply Connected Subsets of the Plane
this is my first question here! Hopefully it is appropriate. Let $\mathbb{A}$ be the punctured plane, i.e. the 'standard' annulus. For compact, connected subsets of the plane (planar continua) $X \...
- 439
21
votes
2
answers
622
views
Morphism from a surface group to a symmetric group, lifted to the braid group
Let $\Sigma_g$ be the fundamental group of the closed orientable surface of genus $g\ge 2$; let $B_n$ be the braid group on $n\ge 3$ braids; let $S_n$ be the symmetric group on $n$ letters; let $p:B_n\...
- 2,479
3
votes
1
answer
403
views
Geometry of the second barycentric subdivision (and Thomason-fibrant replacement)
Is
$\mathrm{sd}^2 (\Delta^n) = \mathrm{sd}^2(\partial \Delta^n) \times \Delta^1 \cup_{\mathrm{sd}^2(\partial \Delta^n) \times \{1\}} Cone(\mathrm{sd}^2(\partial \Delta^n))$
? Here $\mathrm{sd}^2$ ...
- 63.9k
5
votes
0
answers
155
views
Low dimensional homotopy fibration TOP(M) -> TOP(int(M))
In the thesis of Nancy Cardim she proves that for $M$ a topological manifold of dim $\geq 5$ with connected boundary, there exists a homotopy fiber sequence
$C(\partial M)\rightarrow TOP(M) \...
- 93
1
vote
1
answer
240
views
free group actions on a contractible topological space [closed]
Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $W$ be a contractible topological space with a free $\Sigma_k$-action (from the left). Let $X$ be a $CW$-complex and let $X^k$ be the ...
- 1,990
2
votes
1
answer
465
views
induced group actions and covering maps on Eilenberg-Maclane space
Let $M$ be a finite $CW$-complex. Let $\Sigma_k$ be the symmetric group acting on $k$-letters. Suppose there is a free action of $\Sigma_k$ on $M$. Then we have a covering map
$$
f:M\to M/\Sigma_k.
...
- 2,223
5
votes
1
answer
502
views
Are two equivariant maps between aspherical topological spaces homotopic?
Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial higher homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on ...
- 73