All Questions
Tagged with gn.general-topology at.algebraic-topology
565 questions
8
votes
2
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578
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Maximal trivialising subspace for a vector bundle
Let $X$ be a locally compact Hausdorff space. Given a vector bundle $p: E\to X$, a subspace $Y$ of $X$ is called trivialising (for this bundle), if after restricting this bundle to $Y$, it is a ...
- 403
8
votes
3
answers
1k
views
Lifting symmetries to the universal cover
If $X$ is a connected topological space with universal cover $p: \tilde{X} \to X$, I believe any homeomorphism $f : X \to X$ can be 'lifted' to a homeomorphism $\tilde{f} : \tilde{X} \to \tilde{X}$. ...
- 22.2k
8
votes
2
answers
797
views
Morphism with connected fibers induce surjection on fundamental groups?
Let $X,Y$ be path-connected finite CW complexes with base points $x_0,y_0$, let $f\colon X\to Y$ be a surjective continuous map, such that for every $y\in Y$, the fiber $f^{-1}(y)$ is path connected. ...
user39380
8
votes
1
answer
1k
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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
8
votes
2
answers
476
views
A property stronger than the fixed point property
Assume that $X$ is a topological space. We say that $X$ satisfies the strong fixed point property if the graph of every surjective continuous self-map intersect the graph of every continuous self-...
- 356
8
votes
1
answer
581
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About a generalization of the Borsuk-Ulam theorem
I've come upon this MO post, and I was wondering if there is a way to generalize what is said in the answer at the very bottom. Specifically, we have that given continuous maps $f: \mathbb{S}^n \to \...
- 1,763
8
votes
1
answer
768
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What information can one recover from the induced map on homology?
The following question came up while constructing delay embeddings of time series data.
Consider an unknown topological space $X$ and an unknown continuous function $f:X \to X$. We are given a ...
- 15.5k
8
votes
1
answer
716
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Topological fraction rings and fields
Linked to this question
and as a sequel to my answer of it.
Let $R$ be a topological (commutative, unital) ring and set $S$ be a submonoid of $(R,\times,1_R)$.
Let
$$
s_{frac}\ :\ R\times S\to S^{-...
- 3,738
8
votes
1
answer
477
views
Face poset of a subcomplex complement
Let $P$ denote the face poset of a simplicial complex, $\Delta$ the order complex of a poset, and $\sim$ homotopy equivalence. It's known that for any finite simplicial complex $\mathcal{K}$ that $\...
- 191
8
votes
1
answer
1k
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How many ways do we have to prove that a mapping is open?
Given a continuous mapping $f$ between Euclidean domains
(or domains in topological manifolds) of the same (topological) dimension, what are the natural assumptions to conclude that $f$ is open? Here,...
- 1,881
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 ...
- 81
8
votes
0
answers
172
views
The pro-discrete space of quasicomponents of a topological space
Let $X$ be a topological space.
Consider the functor $P^X : \textbf{Set} \to \textbf{Set}$ that sends each set $Y$ to the set of continuous maps $X \to Y$.
It is not hard to check that $P^X : \textbf{...
- 15.8k
8
votes
0
answers
291
views
Loop space functor and sequential colimits of inclusions
The question is about a fact that is mentioned as "evident" everywhere in the literature, so my guess is that some small detail is passing over my head. Here it is:
Let $X_0\hookrightarrow X_1 \...
- 91
7
votes
4
answers
10k
views
Studying topology: which first, algebraic or differential? [closed]
I have recently studying the basics of topology (ideas in point set, connectedness compactness) and I want to continue my studies but i'm interested in both differential and algebraic topology. which ...
- 129
7
votes
1
answer
2k
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Realizing homomorphisms between fundamental groups
Let $X,Y$ be compact connected manifolds and $\varphi\colon\pi_1(X)\to\pi_1(Y)$ be a homomorphism between their fundamental groups. Under what conditions on $X$, $Y$ and $\varphi$ is it true that $\...
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 ...
- 1,177
7
votes
1
answer
801
views
A generalization of the Borsuk Ulam theorem
Is there a compact $n$-dimensional manifold $M$ or, more generaly, a compact $n$-dimensional topological space $M$ with the following property?
"For every continuous map $f:M \to \mathbb{R}^{...
- 356
7
votes
2
answers
473
views
Trivial cobordism group in dimensions 1, 3, 7 related to H-space structures on the spheres in these dimensions?
Is there a connection between the existence of H-space structures on $S^1$, $S^3$ and $S^7$ and the fact that every (closed) 1-manifold, 3-manifold and oriented 7-manifold is a boundary, or is this a ...
- 931
7
votes
1
answer
692
views
Homotopically trivial vs isotopically trivial diffeomorphisms
Let $M$ be a manifold. Let's say $M$ is smooth, connected, oriented. We can also assume that $M$ is closed if that makes things easier.
Let $\mathit{Diff}(M)$ denote the group of diffeomorphisms of $...
- 1,337
7
votes
1
answer
1k
views
A problem on infinite dimensional metric space
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$...
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 ...
- 63.9k
7
votes
1
answer
557
views
Continuous maps $f:S^n \to \mathbb{C}P^m$ with $f(x)\perp f(-x) $
Question 1: What is a complete classification of all positive integers $m,n$ with the following property:
There is a continuous map $f:S^n \to \mathbb{C}P^m$ such that $f$ maps antipodal ...
- 356
7
votes
1
answer
304
views
Does the CGWH-fication change the (weak) homotopy type?
Let $Top$, $CG$, $WH$, $CGWH$ be the categories of topological spaces, compactly generated spaces, weak Hausdorff spaces and compactly generated weak Hausdorff spaces.
There is the CG-ification $X_{...
- 71
7
votes
2
answers
2k
views
Intersection form in twisted homology (homology with local coefficients)
The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local ...
- 1,129
7
votes
1
answer
1k
views
G-equivariant Whitehead's Theorem
Suppose $X$ is a CW complex and $Y$ is a subcomplex. Let $G$ be a compact Lie group that acts on $X$ and $Y$. Suppose further that the CW structures on $X$ and $Y$ are $G$-stable. Moreover assume ...
- 8,529
7
votes
3
answers
911
views
A fibrant-objects structure on Top
(Sorry for the crossposting, but I'm really interested in this question).
One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
- 13.6k
7
votes
2
answers
419
views
Relation between $KO$ and $K$
What can be said about the relation between the complex and the real K-theory of a CW complex? An $n$-dimensional complex vector bundle is an $2n$-dimensional real vector bundle but not vice versa. ...
7
votes
1
answer
397
views
A set theoretic question arising from trying to understand a sheaf cohomology question
I'm trying to understand the footnote to Example 5.3 in Wiegand - Sheaf cohomology of locally compact totally disconnected spaces which is about constructing a locally compact Hausdorff and totally ...
- 38.6k
7
votes
1
answer
614
views
Compactification of open manifolds in the form of a manifold( with zero Euler characteristic)
Edit: According to the interesting comments of Michael Albanese and Nick L we revise the question as follows:
By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ ...
- 356
7
votes
4
answers
1k
views
Quotient rings of $C(X)$
Let $X$ be a Tychonoff topological space. Consider the ring $C(X)$ of all continuous real valued functions on $X$. For what conditions on an ideal $I$ of $C(X)$, we could deduce that the quotient ring ...
- 1,788
7
votes
2
answers
616
views
Which topological spaces contain dense simply connected subspace?
And when can this subspace be chosen to be open?
As the answer to this question indicates, any manifold contains an open dense subset, which is homeomorphic to $\mathbb{R}^{n}$, and so for manifolds ...
- 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
7
votes
1
answer
200
views
Quasifibrations and transfinite filtrations
This question takes place in the category $\mathrm{CGWH}$
of compactly generated weak Hausdorff spaces.
Let $\lambda$ be a limit ordinal, and suppose we have
a diagram $\Phi: \lambda \to \mathrm{CGWH}$...
- 12.5k
7
votes
1
answer
504
views
Topology of connected subsets of the $3$-torus
Consider the $3$-torus $Y=T^3$, a subset $\Sigma\subset Y$, and $\Sigma^*=Y\setminus\overline\Sigma$.
We assume both $\Sigma$ and $\Sigma^*$ to be open, connected, and smoothly bounded.
I am ...
- 181
7
votes
0
answers
158
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 ...
- 759
7
votes
1
answer
353
views
Does the category of cosheaves have enough projectives?
Given a general topological space $X$ does the category $\mathbf{coShv}(X,\mathbf{Mod}_R)$ have enough projectives ? I know that under some conditions this is true, for example if $X$ is a cell ...
- 173
7
votes
0
answers
119
views
The automorphism group of the fibered cylinder
My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that ...
- 41.8k
7
votes
0
answers
287
views
Does geometric realization commute with passing to the compactly generated topology?
My question is in the title, but here is a more detailed formulation:
Let Top be the category of all topological spaces and continuous maps, and let CGTop be the subcategory of compactly generated ...
- 8,803
7
votes
0
answers
430
views
algebraic structure of Integral Steenrod squares
It is well known that the classical Steenrod squares $Sq^a$ satisfy the Adem relations
$$Sq^aSq^b= \sum_c \binom{b-c-1}{a-2c}Sq^{a+b-c}Sq^c\;.$$
In the case where $a$ is odd, one can define an ...
- 928
7
votes
0
answers
570
views
Thom Class of tensor bundles
Suppose $\xi$ and $\eta$ are oriented vector bundles over a CW-complex $B$. Is it possible to express the Thom class (with ${\mathbb Z}$ coefficients) of $\xi\otimes \eta$ or even ${\rm Sym}^2(\xi)$ ...
- 2,830
6
votes
3
answers
770
views
Null-homotopy of diagonal map
For a sphere $S^n$, the diagonal map $\Delta:S^n\to S^n\wedge S^n$ sending $x\mapsto x\wedge x$ is null-homotopic. This is the homotopy group $\pi_n(S^n\wedge S^n)=\pi_n(S^{2n})$ is trivial.
I'm ...
- 681
6
votes
4
answers
926
views
On the homotopy type of $\mathbb{QP}^\infty$
It can be shown that the infinite-dimensional rational projective space $\mathbb{QP}^\infty$ is a connected, Hausdorff topological space. What can be said about its homotopy type (is it simply ...
- 13.6k
6
votes
3
answers
887
views
Finite CW complex with finite non-abelian fundamental group and higher homologies zero
I want to build a finite CW complex such that $\pi_1$ is non-abelian and $H_i$ are zero for $i\geq 2.$
From Hatcher for a given group G, one can create an example of a 2-complex $X_G$ with $\pi_1(X_G)=...
- 1,177
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 ...
- 3,317
6
votes
5
answers
919
views
A question about local connectedness in metric spaces
Must every compact and connected metric space be locally connected at at least one
of its points?
- 6,215
6
votes
1
answer
439
views
Classify $K(\pi,n)$ that are manifolds
Inspired by `Naturally occuring' $K(\pi, n)$ spaces, for $n \geq 2$. and When is a classifying space a topological manifold?, I'd like to formulate a precise question:
For which $n \in \mathbb{Z}...
- 15.4k
6
votes
3
answers
2k
views
Uniquely geodesic and CAT(0) spaces?
Improvement after J-M Schlenker's comment below :
This post has been divided into two parts, the second part is here.
Question : Is a finite dimensional metric space, uniquely geodesic if and only ...
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
6
votes
2
answers
5k
views
On the cohomology of a finite covering map
So let $X$ be a "nice" topological space and assume that $G$ is a finite group which acts freely on $X$.
Q: Is there a simple relationship between the cohomology groups
$H^i(G,\mathbf{Z}), H^i(X,\...
- 7,609
6
votes
2
answers
1k
views
Fundamental groups and homology groups of closed subsets of the plane
Let $X$ be a closed subset of $\mathbb{R}^2$. What restrictions are there on $\pi_1(X)$ and on the homology groups of $X$ (both singular and Cech)? This is elementary if $X$ has reasonable local ...
- 207