All Questions
Tagged with gn.general-topology at.algebraic-topology
565 questions
231
votes
4
answers
16k
views
Is $\mathbb R^3$ the square of some topological space?
The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \...
- 5,275
140
votes
7
answers
34k
views
Is the boundary $\partial S$ analogous to a derivative?
Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after ...
- 151k
76
votes
9
answers
15k
views
understanding Steenrod squares
There is a function on $\mathbb{Z}/2\mathbb{Z}$-cohomology called Steenrod squaring: $Sq^i:H^k(X,\mathbb{Z}/2\mathbb{Z}) \to H^{k+i}(X,\mathbb{Z}/2\mathbb{Z})$. (Coefficient group suppressed from ...
- 6,069
72
votes
9
answers
9k
views
What is a continuous path?
I would like some help, because I am getting mad trying to answer the following
Question: Let $X$ be a topological space, what is a continuous path in $X$?
Well, maybe you're already getting ...
- 6,034
70
votes
28
answers
7k
views
Examples where it's useful to know that a mathematical object belongs to some family of objects
For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
68
votes
3
answers
21k
views
Properly Discontinuous Action
When looking definition, and theorems related to Properly discontinuous action of a group $G$ on a topological space $X$, it is different in different books (Topology and Geometry-Bredon, Complex ...
- 1,236
66
votes
5
answers
8k
views
Does homology have a coproduct?
Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...
- 661
66
votes
4
answers
6k
views
Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
- 1,187
60
votes
6
answers
7k
views
Torsion in homology or fundamental group of subsets of Euclidean 3-space
Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups is not ...
- 44.3k
51
votes
5
answers
9k
views
Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
- 63.1k
48
votes
3
answers
13k
views
When is a Homology Class Represented by a Submanifold? [duplicate]
Possible Duplicate:
Cohomology and fundamental classes
Given an oriented manifold $M$ and an oriented submanifold $\phi:N\to M$ we can obtain a homology class $\phi_*[N]\in H_*(M)$ ...
- 2,283
44
votes
7
answers
22k
views
How do you show that $S^{\infty}$ is contractible?
Here I mean the version with all but finitely many components zero.
- 10.5k
44
votes
6
answers
4k
views
Does $\mathbb C\mathbb P^\infty$ have a group structure?
Does $\mathbb C\mathbb P^\infty$ have a (commutative) group structure? More specifically, is it homeomorphic to $FS^2$, (the connected component of) the free commutative group on $S^2$?
$\mathbb C\...
- 8,717
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
39
votes
3
answers
6k
views
Why do finite homotopy groups imply finite homology groups?
Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\...
- 2,782
38
votes
3
answers
2k
views
If $X$ and $Y$ are homotopy equivalent, then are $X \times \mathbb{R}^{\infty}$ and $Y \times \mathbb{R}^{\infty}$ homeomorphic?
Let $X$ and $Y$ be reasonable spaces. Since $\mathbb{R}^{\infty}$ is contractible,
$$
X \times \mathbb{R}^{\infty} \cong Y \times \mathbb{R}^{\infty} \;\;\; \implies \;\;\; X \simeq Y.
$$
Is the ...
- 5,898
37
votes
1
answer
1k
views
Does there exist a continuous 2-to-1 function from the sphere to itself?
I am interested in the following question:
Does there exist a continuous function $f:S^2\to S^2$ such that, for any $p\in S^2$, $|f^{-1}(\{p\})|=2$?
I suspect the answer is no, but I don't know ...
- 513
36
votes
3
answers
6k
views
In a topological space if there exists a loop that cannot be contracted to a point does there exist a simple loop that cannot be contracted also?
I'm interested in whether one only needs to consider simple loops when proving results about simply connected spaces.
If it is true that:
In a Topological Space, if there exists a loop that cannot ...
- 4,862
36
votes
4
answers
5k
views
Compact open topology on $\mathrm{Homeo}(X)$
Let $X$ and $Y$ be topological spaces. Define the compact open topology on the set $\mathrm{M}(X,Y)$ of continuous maps from $X$ to $Y$ via the subbase $[K,O]$ of all maps $f:X\rightarrow Y$ s.t. $f(K)...
- 2,751
35
votes
2
answers
5k
views
Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"?
This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).
The weak Hausdorff rather ...
- 2,974
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
34
votes
6
answers
4k
views
Why study finite topological spaces?
In rereading Thurston's essay On Proof and Progress in Mathematics I ran across this passage:
… this means that some concepts that I use freely and naturally in
my personal thinking are foreign to ...
- 737
33
votes
2
answers
2k
views
What happened to the last work Gaunce Lewis was doing when he died?
In 2006, Gaunce Lewis died at the age of 56. He'd done important work setting up equivariant stable homotopy theory, and I think it's fair to say his work was far ahead of its time. In recent years, ...
- 30.3k
31
votes
3
answers
1k
views
Non embedding of $Y\times Y$ into $\mathbb{R}^3$
I know that this is a well known result, but where can I find a proof? I am also interested to see more general non-embedding results of this type.
Theorem. Let $Y$ be the union of two segments ...
- 28k
30
votes
5
answers
4k
views
The role of ANR in modern topology
Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ ...
30
votes
2
answers
2k
views
Does there exist any non-contractible manifold with fixed point property?
Does there exist any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that the ...
- 3,751
30
votes
5
answers
2k
views
Is the universal covering of an open subset of $\mathbb{R}^n$ diffeomorphic to an open subset of $\mathbb{R}^n$ ?
Is the universal covering of a connected open subset $U$ of ℝn diffeomorphic to an open subset of ℝn (standard differentiable structure)?
If not true in general, is there any condition ...
28
votes
5
answers
4k
views
Two-to-one continuous mapping from R² to R²
Hello. I have a question.
Does there exist a continuous mapping
$F:\mathbb{R}^2\rightarrow\mathbb{R}^2$
such that for every $c\in F(\mathbb{R}^2)$
there are two and only two points $z_{1}$, $z_{2}$...
- 301
27
votes
6
answers
3k
views
Applications of string topology structure
Chas and Sullivan constructed in 1999 a Batalin-Vilkovisky algebra structure on the shifted homology of the loop space of a manifold: $\mathbb{H}_*(LM) := H_{*+d}(LM;\mathbb{Q})$. This structure ...
- 8,167
27
votes
1
answer
4k
views
connectivity of the group of orientation-preserving homeomorphisms of the sphere
In the paper "Local Contractions and a Theorem of Poincare" Sternberg has mentioned the following question which was open when the paper was written:
Is the group of orientation-preserving ...
- 6,214
26
votes
3
answers
1k
views
Proving that a function's image contains (1/n,...,1/n)
This question is a follow-up to a previous question answered by Neil Strickland:
Map from simplex to itself that preserves sub-simplices
Let $B$ denote the closed unit ball in $\mathbb{R}^2$ and let ...
- 645
26
votes
1
answer
1k
views
Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
- 27.5k
25
votes
6
answers
5k
views
Is there a classification of open subsets of euclidean space up to homeomorphism?
I hope this question is reasonable enough to have a well known answer. i.e either there is a simple invariant (like the homotopy groups) that characterizes the homeomorphism type of such set among ...
- 2,027
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
25
votes
1
answer
5k
views
Example of fiber bundle that is not a fibration
It is well-known that a fiber bundle under some mild hypothesis is a fibration, but I don't know any examples of fiber bundles which aren't (Hurewicz) fibrations (they should be weird examples, I ...
- 253
25
votes
2
answers
2k
views
CW complexes and paracompactness
It seems like when we assume "niceness" in homotopy theory we assume that $X$ has the homotopy type of a CW complex, and in fiber bundle theory we assume that $X$ is paracompact. How do these two ...
- 1,207
24
votes
2
answers
4k
views
complement of a totally disconnected closed set in the plane
While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
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,...
- 956
23
votes
5
answers
2k
views
The "right" topological spaces
The following quote is found in the (~1969) book of Saunders MacLane,
"Categories for the working mathematician"
"All told, this suggests that in Top we have been studying
the wrong mathematical ...
- 18.7k
23
votes
1
answer
1k
views
Which spaces have the (weak) homotopy type of compact Hausdorff spaces?
Inspired by the discussion in the comments of this question, I'd like to ask the following question: is it possible to characterize the class of spaces that are homotopy equivalent (or weak equivalent)...
- 31.2k
22
votes
0
answers
676
views
Are there "chain complexes" and "homology groups" taking values in pairs of topological spaces?
Throughout this question, notation of the form $(X,A)$ denotes a sufficiently nice pair of topological spaces. I think for most of what I'm saying here, it is enough to assume that the inclusion $A \...
- 15.5k
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 ...
21
votes
5
answers
1k
views
Explanation for E_8's torsion
To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is ...
- 15.1k
20
votes
2
answers
1k
views
The Gelfand duality for pro-$C^*$-algebras
The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
- 1,344
19
votes
4
answers
4k
views
When is a finite cw-complex a compact topological manifold?
I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-...
- 732
18
votes
3
answers
2k
views
Are finite spaces a model for finite CW-complexes?
Are finite topological spaces (i.e. topological spaces whose underlying set is finite) a model for the homotopy theory of finite simplicial sets (= homotopy theory of finite CW-complexes) ?
Namely, ...
- 43.2k
18
votes
3
answers
3k
views
What are parabolic bundles good for?
The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate ...
- 2,751
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
17
votes
3
answers
954
views
Can an oriented closed $n(\geq 2)$-dimensional manifold be smoothly embedded in $\mathbb{R}^{2n-1}$?
Can anyone provide me with an example of an orientable closed manifold $M$ of dimension $n\geq 2$, which cannot be smoothly embedded in $\mathbb R^{2n-1}$?
I know these cannot exist for $n=1$, i.e. $S^...
- 3,751
17
votes
8
answers
3k
views
Smooth classifying spaces?
Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...
- 1,422