All Questions
116 questions
10
votes
1
answer
659
views
Are there any tests for knowing whether a topological space admits a CW structure?
We know that for n $\ge$ 5, a manifold admits a piecewise linear structure if and only if its Kirby-Siebenmann class vanish and Galewski and Stern showed the existence of a similar invariant to test ...
- 168
5
votes
1
answer
380
views
Proving the Cork Theorem
I am reading Kirby's paper paper "Akbulut's corks and h-cobordisms of smooth simply connected 4-manifolds" and I have a question about how to actually prove the cork theorem from the results ...
14
votes
0
answers
326
views
When can we extend a diffeomorphism from a surface to its neighborhood as identity?
Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
- 3,751
1
vote
0
answers
48
views
Connected pre-images spanning $n$-cubes under dimension reducing maps
Let $I^n = [0,1]^n$ be the $n$-dimensional hypercube. For a continuous function $f: I^n \to \mathbb{R}^m$ with $m < n$, we're interested in the existence of points $p \in \mathbb{R}^m$ whose ...
user530766
0
votes
1
answer
135
views
Local embedding and disk in domain perturbation
Consider say $M=(\mathbb{S}^1\times\dotsb\times \mathbb{S}^1)-q$ ($n$-times). Assume that $B$ is an $n$ disk in $M$ (for instance, thinking of $\mathbb{S}^1$ as gluing $-1$ and $1$, the cube $B=[-\...
1
vote
0
answers
61
views
Necessary or sufficient conditions for the $k$-fold intersection to be empty in a covering with a "tree structure"
Consider a finite collection of $d$-dimensional balls $\mathfrak{B}=\{B_1,\ldots,B_n\}$ which cover a PL $d$-manifold $M$, i.e. $M=\bigcup_{i=1}^{n}B_i$. Suppose we want to compute the Euler ...
- 159
0
votes
1
answer
328
views
Relationship between quotient CW-complexes after attaching cells
I have been trying to prove the following simple-looking result which I require for some work in low-dimensional topology. I expect it is likely true and in a textbook somewhere so any reference or ...
- 187
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
$$
\left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
- 705
1
vote
0
answers
145
views
Determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
What and how to determine the Eilenberg-MacLane spaces on the right-handed side of this Whitehead tower?
Namely, how do we know
$$
K(Z_2,1)?, \quad K(Z_2,2)?, \quad K(Z,4)?
$$
Naively -- in each step ...
- 447
5
votes
0
answers
249
views
Aspherical space whose fundamental group is subgroup of the Euclidean isometry group
Let $M$ be a smooth, compact manifold without a boundary, with its universal covering $\tilde{M} = \mathbb{R}^n$. If there exists an injective homomorphism $h: \pi_1(M) \rightarrow O(k) \ltimes \...
- 661
3
votes
1
answer
260
views
Can such a set be simply connected?
$\newcommand\R{\mathbb R}$Let $U$ be an open subset of $\R^2$ such that the point $(0,0)$ is on the boundary of $U$. Let $f\colon[0,1]\to\R^2$ be the path that starts at $(0,0)$ and moves with a (say) ...
- 128k
5
votes
2
answers
711
views
On the boundary of a simply connected set
Let $U$ be an open simply connected subset of $\mathbb R^2$. Let $x$ be a boundary point of $U$.
Does then there always exist a continuous function $f\colon[0,1]\to\mathbb R^2\setminus U$ such that $x ...
- 128k
4
votes
0
answers
350
views
Does a contractible locally connected continuum have an fixed point property?
I'm surprised that I can't find any research on this topic. Maybe it's too obvious? Kinoshita proved that contractible continuum do not have FPP, but his example is not locally connected. Maybe if we ...
1
vote
1
answer
177
views
Identifying a curve on a closed surface of genus 4
The notation is the one used in the attached picture.
Take a closed, orientable surface $\Sigma_4$ of genus $4$, obtained as the identification space of a polygon with $16$ sides in the usual way. The ...
- 66.3k
3
votes
0
answers
429
views
"Maehara-style" proof of Jordan-Schoenflies theorem?
The highest upvoted answer to this old question Nice proof of the Jordan curve theorem? is a proof by Ryuji Maehara. I personally really liked/appreciated that Maehara's proof is
A) a fairly ...
- 831
3
votes
1
answer
375
views
Boundaries of subsets of simply-connected domains
I am trying to prove the following assertion: Let $B$ be a simply-connected set, and let $B' \subsetneq B$ be a proper open connected subset. Then, there exists a point $x \in \partial B'$ of the ...
3
votes
1
answer
171
views
Spaces satisfying a strong Cartan-Hadamard theorem
Let $(X,d)$ be a connected geodesic metric space. When does there there exists a covering map $\pi:H\rightarrow X$ which is a local-isometry where $H$ is either a Hilbert space or a Euclidean space?
...
- 364
2
votes
1
answer
130
views
Gluing isotopic smoothings
Let $M$ be a topological manifold which can be written as $M = U \cup V$ where $U$ and $V$ are open. Suppose both $U$ and $V$ admit smooth structures. Also assume that on the overlap $U \cap V$ the ...
- 803
1
vote
1
answer
190
views
Approximations by compact sub-spaces
Suppose $X$ is a Hausdorff (I'm happy to also assume "non compact") topological space that can be written as the topological direct limit
$$\varinjlim_{a\in J} K_a$$
for $J$ a directed set ...
user335418
1
vote
0
answers
143
views
End space of non-compact 2-manifolds described with proper rays
I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. I asked ...
14
votes
3
answers
1k
views
Quotient of solid torus by swapping coordinates on boundary
Let $T$ be the solid 2-torus and let $\sim$ be the equivalence relation on $T$ generated by the relation $\{(\alpha,\beta) \sim (\beta,\alpha) \mid \alpha, \beta \in S^1\}$ on the boundary $\partial T=...
- 1,813
0
votes
0
answers
74
views
Do adjoining basepoints and/or moduli of spaces affect fixed points nicely?
My question is when will $(X_+)^G$ or $(X/A)^G$ be equal to $(X^G)_+$ or $X^G/A^G$ respectively for $X$ a $G$-space, $G$ a finite cyclic group and $X^G$ the ordinary fixed points. These seem like they ...
- 9
40
votes
2
answers
2k
views
Can the nth projective space be covered by n charts?
That is, is there an open cover of $\mathbb{R}P^n$ by $n$ sets homeomorphic to $\mathbb{R}^n$?
I came up with this question a few years ago and I´ve thought about it from time to time, but I haven´t ...
- 10.6k
0
votes
1
answer
271
views
When do cobordism groups depend on differential structure? [closed]
I heard that cobordism group with structures sometimes depend on differential structure of space.
Do you know any examples or references about this facts?
I want to know when difference occur between ...
- 1
13
votes
3
answers
2k
views
A quotient space of complex projective space
Let $\mathbb{C}P^n$ be the $n$-dimensional complex projective space and denote $[z_0:\dots:z_n]$ its points. If we glue $[z_0:\dots:z_n]$ and $[\overline{z_0}:\dots:\overline{z_n}]$ for any $[z_0:\...
- 331
18
votes
0
answers
1k
views
What is the strongest nerve lemma?
The most basic nerve lemma can be found as Corollary 4G.3 in Hatcher's Algebraic Topology:
If $\mathcal U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of ...
- 81
3
votes
0
answers
282
views
Commutator length of the fundamental group of some grope
A popular way to describe a grope as the direct limit $L$ of a nested sequence of compact 2-dimensional polyhedra
$L_0 \to L_1 \to L_2 \to \cdots$
obtained as follows. Take $L_0$ as some $S_g$, an ...
- 2,083
4
votes
1
answer
522
views
Can every manifold be represented as a quotient
My question is "inspired" by the uniformization theorem for Riemmannian surfaces and this post.
Suppose that $X$ is connected (finite-dimensional) topological manifold without boundary. ...
- 5,405
1
vote
1
answer
102
views
Approximate Jordan-Brouwer theorem (corrected)
My first attempt to ask this question sort of failed (I'll explain below).
This came up when thinking about this question.
It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{...
- 5,529
2
votes
1
answer
131
views
Approximate Jordan-Brouwer theorem
This came up when thinking about this question.
It is well-known that the image of a homeomorphic embedding $f:S^n\to \mathbb{R}^{n+1}$ separates the space into exactly two components, one of which is ...
- 5,529
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
25
votes
1
answer
1k
views
Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?
I'm interested in both version of the question in the title, i.e. in the topological category and in the smooth category. By a topological immersion I mean a local embedding. I was asking in ...
- 439
9
votes
1
answer
320
views
Isotopies, Fiber Bundles and Selection Theorems
The following problem is a culmination of a few questions I've asked the last two months, and it's still giving me some issues. I think I know the right way to solve it, but I'm having trouble with ...
- 439
1
vote
0
answers
297
views
Boundary map in Mayer-Vietoris sequence of cohomology
Suppose $M$ is a 3-manifold with connected boundary. Let $T$ be a tangle in $M$, i.e., $T$ is a embedded connected 1-submanifold whose boundary is on $\partial M$ ($T$ is not closed). Moreover, ...
- 673
6
votes
2
answers
324
views
Nonvanishing section of infinite-dimensional tautological bundle
Let $H$ be a real or complex Hilbert space. In the case where $H$ is infinite-dimensional, let us define a half-dimensional subspace as a subspace $W \subset H$ such that both $W$ and $W^\perp$ have ...
- 9,853
1
vote
0
answers
80
views
A characterization for a space that is similar to locally connected spaces
Let $X$ be a $T_0$ topological space with the property that there exists a basis $\{O_i\}_{i\in I}$ for $X$ such that for each $J\subseteq I$ the subspace $\bigcap_{i\in J}O_i$ has only finitely many ...
1
vote
1
answer
423
views
Extension of homeomorphisms
Let $f,g:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be smooth injective and let $n\leq m$. Let $k \in \mathbb{N}$, and let $\iota_m^{m+k}:\mathbb{R}^m\rightarrow \mathbb{R}^{m+k}$ be the canonical ...
- 5,405
2
votes
0
answers
323
views
Continuous injective functions with dense image
Let $X$ be the set of continuous, injective functions from $\mathbb{R}^n$ to $\mathbb{R}^n$ with dense image; and equip $X$ with the (relative) compact-open topology. What is known about this space?
...
- 5,405
10
votes
2
answers
451
views
Group of surface homeomorphisms is locally path-connected
I think the following is true and I need a reference for the proof. (Given a closed surface $S$, i.e. a compact 2-dimensional topological manifold (without boundary), we endow $S$ with a distance ...
- 1,517
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
35
votes
4
answers
4k
views
An intelligent ant living on a torus or sphere – Does it have a universal way to find out?
I wanted to ask a question about topological invariants and whether they are connected in a fundamental or universal way. I am not an expert in topology, so please let me ask this question by way of a ...
- 6,917
1
vote
1
answer
208
views
Is every homeomorphism approximately a product of homeomorphisms?
Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi_1,\phi_2$ on $\mathbb{R}^...
- 5,405
4
votes
0
answers
129
views
'Monodromy' for relative homology group
Let $A$ and $X$ be topological manifolds. Denote by $\mathbb {Emb}(A,X)$ the space of all topological embeddings $A\to X$.
A loop $f_s:A\to X$ ($s\in[0,1]$) in $\mathbb{Emb}(A,X)$ should give rise to ...
- 2,789
21
votes
7
answers
1k
views
Reference for topological graph theory (research / problem-oriented)
I would be interested in recommendations for topological graph theory texts. I think Gross and Yellen has a great chapter on topological graph theory, and I find Mohar and Thomassen's Graphs on ...
4
votes
0
answers
228
views
Enlarging a compact set in order to improve its shape
In my previous question it was established that if $X$ is a metrizable, connected, locally path connected space and $K\subset X$ is compact, then there is a Peano continuum $L\subset X$ such that $K\...
- 5,529
7
votes
1
answer
229
views
Retracting off a compact set
Let $K$ be a compact set in $\mathbb{R}^n$ and let $U$ be a bounded open set that contains $K$. You may assume both are connected.
Can we always find an open $V$ such that $K\subset V\subset\...
- 5,529
1
vote
0
answers
53
views
Spaces that are comparable with their compacts
This is an outgrowth of this question.
For a (metrizable) space $X$ consider the following increasingly strong properties:
(i) For every compact $K\subset X$ there is a map $f:X\to X$ such that $K\...
- 5,529