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7 votes
0 answers
155 views

Maps with small fibers between manifolds of equal dimension

The following question is an attempt to revise this one into what I intended. Important revisions are shown in bold. Are there any known examples of a compact Riemannian manifold $M$ with (possibly ...
Matthew Kvalheim's user avatar
5 votes
1 answer
275 views

Codimension zero embeddings and maps with small fibers

Edit: as explained in my comment on alesia's answer, I mistakenly did not ask below the question I intended (due to my misguided efforts to simplify it). Thus, I revised and reposted my question here. ...
Matthew Kvalheim's user avatar
10 votes
1 answer
654 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 ...
Tyrannosaurus's user avatar
3 votes
2 answers
339 views

Cohomology version of Moore space

I asked this question on MSE a few days back but could not get any helpful response. So I am rewriting the post. It is known to me that given a simply connected finite dimensional (which is also level-...
piper1967's user avatar
  • 1,177
7 votes
2 answers
383 views

Connectivity of fibers under fibration replacement

Assume all the spaces mentioned below are simply connected CW complexes. Let $ f: X \to Y $ be a continuous surjctive map between CW complexes, where $ f $ is not necessarily a fibration. Assume that ...
piper1967's user avatar
  • 1,177
3 votes
1 answer
350 views

How to define relative orientation in terms of (co)homology?

Let $f\colon X\to Y$ be a smooth surjective map of smooth manifolds of dimension $n$ which are not necessarily orientable. A relative orientation of $X$ over $Y$ consists of an isomorphism $\psi\colon ...
Hans's user avatar
  • 3,031
1 vote
0 answers
111 views

Unique Hausdorff topology on trivial vector bundle?

Question: Is there a Hausdorff topology other than the product topology on $X\times \mathbb{C}^n$, that turns $(X\times \mathbb{C}^n, \mathrm{pr}_1)$ into a vector bundle, where $\mathrm{pr_1}$ ...
PHmath's user avatar
  • 11
1 vote
0 answers
37 views

Asymptotic growth of twisted alexander polynomials and hyperbolic volume for infinite families of knots

Let $\{K_n\}_{n=1}^\infty$ be an infinite family of hyperbolic knots with increasing crossing number, and let $\rho_n: \pi_1(S^3 \setminus K_n) \to SL_N(\mathbb{C})$ be a sequence of irreducible ...
Chandler Halderson's user avatar
2 votes
0 answers
146 views

Two open contractible subset of $\mathbb{R}^3$ which are not homeomorphic but are polynomial preimage of open sets of $\mathbb{R}$

About 2 decades ago I heared from some one that there are infinitely many open contractible subsets of space mutually non homeomorphic to each other. I confess that I do not remember the ...
Ali Taghavi's user avatar
6 votes
1 answer
360 views

On connected sum of compact manifolds along a submanifold

Let $M_1$ and $M_2$ be two compact manifolds of dimension $n\ge 3$. Let us have embeddings $i_1: K \to M_1$ and $i_2: K \to M_2$ for a closed manifold $K$ of dimension at most $n-1$ such that the ...
Katrina's user avatar
  • 506
5 votes
1 answer
379 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 ...
failedentertainment's user avatar
1 vote
1 answer
132 views

Is the product of torus and sphere a cover of the symmetric square of torus?

Let $T$ denote the $2$-dimensional torus and $T^{(2)}$ denote its symmetric square (which is the orbit space of the canonical $\mathbb{Z}_2$ action on the $4$-torus $T \times T$). One can see $T^{(2)}$...
SRhonda's user avatar
  • 31
4 votes
1 answer
183 views

When can a generalized connected sum be aspherical

Let $M$ and $N$ be compact $n$-dimensional manifolds with a common (nicely embedded) compact submanifold $S$ (we may assume that the inclusion of $S$ in $M$ and $N$ is $\pi_1$-injective). Let $M\#_S N$...
Jeremy's user avatar
  • 311
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 \...
Anubhav Mukherjee's user avatar
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 ...
user avatar
4 votes
0 answers
106 views

Reference request for a theorem of Jaworowski

Jan Jaworowski, in 2000, proved the following theorem (I came to know about it from here) Jaworowski (2000) : Let $Y$ be a finite simplicial complex of dimension $k$ and let $n\ge 2k$. If $f:S^n\to Y$...
HackR's user avatar
  • 141
5 votes
1 answer
244 views

Does a "good" homotopy equivalence between pairs imply homotopy equivalence between quotient spaces?

If $(X,A)$ and $(Y,B)$ are (good) pairs of topological spaces, and $f:X\rightarrow Y$ is a homotopy equivalence such that the restriction $f\restriction_A$ is a homotopy equivalence between $A$ and $B$...
Ondrej Draganov's user avatar
2 votes
1 answer
174 views

A topological space has the homotopy-type of a CW-complex, then is it locally contractible?

Let $X$ be a topological space which has the homotopy-type of a CW-complex. As well-known, a CW-complex is locally contractible. Question: Is $X$ locally contractible? If not, can some one give a ...
Lelong  Wang's user avatar
0 votes
0 answers
64 views

Can an upper hemicontinuous correspondence be discountinuous everywhere?

Let $\phi: X \rightrightarrows Y$ be an upper hemicontinuous correspondence. If $K \subset X$ is a compact and convex set, $K$ contains an open set $U$, and $\phi(x)$ is nonempty, compact, and convex ...
Kai's user avatar
  • 101
2 votes
0 answers
156 views

Testing for weak homotopy equivalences with compact Hausdorff spaces

Let $f \colon X \to Y$ be a weak homotopy equivalence between topological spaces. If I am not mistaken, then one can rephrase this by stating that the induced map $[K,X] \to [K,Y]$ between homotopy ...
AlexE's user avatar
  • 2,998
8 votes
2 answers
710 views

Could there be any homotopy group without "Lebesgue Number Lemma"?

This is about a comment that I have made in my general topology class while I was proving the abovementioned lemma as a consequence of compactness! As far as I know, essentially, there is only one ...
user51223's user avatar
  • 3,173
1 vote
0 answers
177 views

If $X$ is a strong deformation retract of $\mathbb{R}^n$, then is $X$ simply connected at infinity?

Let $X \subseteq \mathbb{R}^n$, and assume there is a strong deformation retract from $\mathbb{R}^n$ to $X$. Is $X$ necessarily simply connected at infinity? (Edit) Follow up question: if there is a ...
ccriscitiello's user avatar
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=[-\...
monoidaltransform's user avatar
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 ...
rab's user avatar
  • 159
2 votes
0 answers
55 views

Fundamental group of cyclic branched cover of affine plane

Let $f\in \mathbb{C}[x,y]$ be an irreducible polynomial. Let $n>0$ be an integer such that the hypersurface $S:=\{ (x,y,z)\in \mathbb{C}^3|z^n=f(x,y) \}$ is a connected complex submanifold of $\...
Doug Liu's user avatar
  • 615
7 votes
2 answers
209 views

Cover the $n$-disc irredundantly with $n+1$ open sets. Suppose that the $(n+1)$-fold intersection is empty. Then is some $n$-fold intersection empty?

Cover the $n$-disc with $n+1$ open sets $D^n = U_0 \cup \dotsb \cup U_n$. Suppose that $U_0 \cap \dotsb \cap U_n = \emptyset$. Suppose moreover that the cover is irredundant in the sense that no ...
Tim Campion's user avatar
  • 63.9k
0 votes
1 answer
327 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 ...
William Thomas's user avatar
12 votes
1 answer
379 views

Approximate classifying space by boundaryless manifolds?

As pointed out by Achim Krause, any finite CW complex is homotopy equivalent to a manifold with boundary (by embedding into $\mathbb R^n$ and thickening), and so every finite type CW complex can be ...
0207's user avatar
  • 123
0 votes
0 answers
150 views

Connectedness of deleted symmetric product

Let $X$ be a connected Hausdorff space. It is well-known that the $n$-fold symmetric product $\mathcal{F}_n(X) := \{A\subseteq X : 0<|A|\leq n\}$ is a connected space equipped with the Vietoris ...
Peluso's user avatar
  • 674
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(...
Ola Sande's user avatar
  • 705
2 votes
1 answer
179 views

Factorization systems for vector bundles

Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
Siya's user avatar
  • 615
0 votes
0 answers
100 views

Finding an example if it exists, for a non-contractible and contractible space with special requirement on quotients of their union?

Let $A$ and $B$ be subsets of $n$-dimensional Euclidean space $\mathbb{R}^{n}$, such that $A$ is non-contractible, $B$ is contractible and $B$ is not an one-point set. I would like to find example(s) ...
Himanshu Yadav's user avatar
5 votes
1 answer
270 views

Are Euclidean spaces $\Delta$-generated?

From the definition of $\Delta$-generated it seems like $\mathbb R$ should be $\Delta$-generated, as $\mathbb R$ is final with respect to all continuous maps $\mathbb R^n \to \mathbb R$. However, the ...
William B.'s user avatar
24 votes
1 answer
1k views

What topological principle is at work here?

[I'm cross-posting this from MSE. I initially asked there 10 days ago, and the question was well-received, but left unanswered.] My question is inspired by a problem I discovered in Putnam and Beyond,...
Yly's user avatar
  • 956
4 votes
0 answers
249 views

Homotopy group of maps into S^3 using its Lie group multiplication to define the group operation

The Bruschlinsky group of maps of a space X into S1 up to homotopy, using the multiplication on S1, is well-known to equal the first cohomology group of X (at least assuming X is a reasonably nice ...
Daniel Asimov's user avatar
6 votes
1 answer
231 views

Weakly contractible $X$, but none of the maps $*\to X$ are cofibrations

Let $\mathrm{Top}$ be the category of all topological spaces and continuous maps. The Quillen model structure on $\mathrm{Top}$ has weak equvalences $W = \{ \text{weak homotopy equivalences} \}$, ...
mathmo's user avatar
  • 331
1 vote
1 answer
153 views

For topological torus action, there is a subcircle whose fixed point is the same as the torus

Let $T=\mathbb{S}^{1}\times \mathbb{S}^{1}\times \cdots \times \mathbb{S}^{1} $ ($n$ times) be an $n$-dimensional torus acting on any topological space $X$. The group $G$ is said to act on a space $X$ ...
Mehmet Onat's user avatar
  • 1,367
2 votes
0 answers
164 views

Triviality of map $(\Sigma \theta)^*$

We know that there is a cofibration sequence $$S^{4n+1}\xrightarrow{\theta}\Sigma^{4m-1} Q_{n-m} \rightarrow \Sigma^{4m-1} Q_{n-m+1} \rightarrow S^{4n+2}\xrightarrow{\Sigma\theta}\Sigma^{4m} Q_{n-m}.$$...
Sajjad Mohammadi's user avatar
3 votes
1 answer
203 views

Simple closed curves in a simply connected domain

Let $U$ be a bounded simply connected domain in the plane. Let $K$ be the boundary (or frontier) of $U$. For every $\varepsilon>0$ is there a simple closed curve $S\subset U$ such that the ...
D.S. Lipham's user avatar
  • 3,317
6 votes
5 answers
953 views

Two arcs in the complement of a disc must intersect?

Let $D=\{z\in \mathbb C:|z|\leq 1\}$ be the unit disc in the complex plane, with interior $U=\{z\in \mathbb C:|z|<1\}$. Let $A\subset \mathbb C\setminus U$ be an arc intersecting $D$ only at its ...
D.S. Lipham's user avatar
  • 3,317
4 votes
2 answers
292 views

$\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ or a correction to this quotient group relation

We know that there is a fiber sequence: $$ \dotsb \to B^3 \mathbb Z \to B \mathrm{String} \to B \mathrm{Spin} \to B^4 \mathbb Z \to \dotsb. $$ Is this fiber sequence induced from a short exact ...
zeta's user avatar
  • 447
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 ...
zeta's user avatar
  • 447
1 vote
0 answers
237 views

Examples of when $X$ is homotopy equivalent to $X\times X$

I was thinking about this question the other day: When is a topological space $X$ homotopy equivalent to $X\times X$ (with the product topology)? This is essentially a cross-post of this MSE question.....
pyridoxal_trigeminus's user avatar
2 votes
0 answers
92 views

Explicit CW-complex replacement of the space of reparametrization maps

Let $P$ be the space of nondecreasing surjective maps from $[0,1]$ to itself equipped with the compact-open topology: $P$ is contractible. There exists a trivial fibration $P^{cof} \to P$ from a CW-...
Philippe Gaucher's user avatar
2 votes
0 answers
103 views

Unordered configuration space with non-distinct points

Consider a topological space $X$, a natural number $n>0$ and the quotient topological space $Q_n(X)$ of $X^n$ by the equivalence relation : $x\sim y$ if and only if there is a permutation $\sigma$ ...
Phil-W's user avatar
  • 1,035
8 votes
0 answers
225 views

A variation of necklace splitting

Our problem is the following: Let $n$ and $k$ be integers. We are given two (unclasped) necklaces, each with $n$ colored stones: a top necklace which has $k$ colors and a bottom necklace which has 2 ...
Sam King's user avatar
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 \...
Chicken feed's user avatar
3 votes
2 answers
285 views

Cut a homotopy in two via a "frontier"

Consider a space $G$ obtained by glueing two disjoint cobordisms (the fact that they are might be irrelevant, assume they are topological spaces at first) $L$ and $R$ on a common boundary $C$. (...
Valentin Maestracci 's user avatar
1 vote
0 answers
130 views

Can we construct a general counterexample to support the weak whitney embedding theorm?

The weak Whitney embedding theorem states that any continuous function from an $n$-dimensional manifold to an $m$-dimensional manifold may be approximated by a smooth embedding provided $m > 2n$. ...
li ang Duan's user avatar
0 votes
0 answers
161 views

Gluing faces of n-cube

Assuming $C_n$ be the $n$-cube, the intersection of $C_n$ with a supporting hyperplane $H(P, v)$ is called a face or more precisely a $d$-face if the dimension is $d$. Let $f_0$ and $f_1$ be faces ...
mahu's user avatar
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