Skip to main content

All Questions

Filter by
Sorted by
Tagged with
8 votes
2 answers
793 views

Does there exist a Haken manifold where all its incompressible surfaces are non-separating?

Every non-zero element in $H_2(M,\mathbb Z)$ corresponds to an incompressible surface. So these surfaces are non-separating. But I'm interested in knowing about separating incompressible surfaces. A ...
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 ...
8 votes
1 answer
3k views

Homotopy equivalence from contractibility of fiber

Suppose $X$ and $Y$ are two $CW$ complexes and $f:X\rightarrow Y$ is a continuous surjection such that fiber of each point (i.e. $f^{-1}(y)$ for each $y\in Y$) is contractible. Does it implies that $...
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 ...
0 votes
1 answer
129 views

Fundamental Surfaces in 3-manifolds

Given a 3-manifold $M$ with a triangulation $T$, will every essential surface in $M$ be a fundamental one? If not, then what are the conditions on $T$ so that these essential surfaces become ...
7 votes
0 answers
504 views

Intersection form of logarithmic transformations

Now I want to calculate the intersection form of a logarithmic transformation which is defined as follows. Let $X$ be an oriented, closed, simply-connected 4-manifold and $T^2\subset X$ be an ...
2 votes
1 answer
226 views

Moving chord on the simple closed curve

Consider a simple closed curve $C$ in $\mathbb{R}^2$. For any points $a$ and $b$ on this curve we associate point $c$ on the left (or right) side to chord $ab$ such that $\angle acb = 90^{\circ}, ac=...
10 votes
3 answers
873 views

Surface Eversions: Generalizing from Sphere and Torus Eversions

In 1958, Smale proved that a $2$-sphere can be "turned inside out", and throughout the 60s, 70s, and 80s, explicit constructions such as Thurston Corrugations, and Minimax eversions were developed to ...
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$^4$, ...
5 votes
1 answer
258 views

Generating the topology of a manifold

Let $X$ be a topological manifold of dimension $d$, and let $F$ be a collection of continuous maps from $X$ into $\mathbf{R}^d$ such that: $F$ separates points of $X$, i.e. for any two distinct ...
16 votes
2 answers
820 views

Klee's trick --- more applications

In his "Some topological properties..." (1955), Klee gave a construction (simple and beautiful) of an isotopy $h_t\colon\mathbb{R}^{2\cdot n}\to \mathbb{R}^{2\cdot n}$ which moves any compact set $K$ ...
17 votes
1 answer
574 views

Simply connected slices

Assume $\Omega$ is an open set in $\mathbb R^3$ such that the intersection of $\Omega$ with any horizontal plane is simply connected. Can you prove that $\Omega$ is simply connected? (Note that ...
1 vote
0 answers
105 views

The inverse image of a Noetherian topological space

A topological space $X$ is called Noetherian if closed subsets satisfy the descending chain condition, equivalently, the open subsets satisfy the ascending chain condition. Let $A$ and $B$ be ...
5 votes
1 answer
192 views

Equivariant cohomology defined by restrictions?

Suppose that $G=S^1$ acts on a smooth, connected, compact manifold with discrete fixed points, additionally assume that there is at least one fixed point. Let $\alpha \in H^{2}_{S^1}(M)$ be such ...
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 ...
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 $n$-...
49 votes
3 answers
8k views

Thurston's 24 questions: All settled?

Thurston's 1982 article on three-dimensional manifolds1 ends with $24$ "open questions":       $\cdots$ Two naive questions from an outsider: (1) Have all $24$ now been resolved? (2)...
2 votes
1 answer
130 views

Are negatively curved $2$-complexes homeomorphic to quotients of the form $\mathbb{H}^{2}/G$, where $G$ is some group?

Let $\mathbb{H}^{2}$ be hyperbolic $2$-space and $G$ a discrete subgroup of $PSL_{2}(\mathbb{R})$. Then the quotient $\mathbb{H}^{2} / G$ is a surface. Can this be extended to $2$-complexes? In other ...
16 votes
4 answers
2k views

Self-covering spaces

Let $M$ be a connected Hausdorff second countable topological space. I will call $M$ self-covering if it is its own $n$-fold cover for some $n>1$. For instance, the circle is its own double cover ...
2 votes
1 answer
308 views

Does any smooth orbifold can be triangulated by orbi-simplex(triangulation of orbifolds)

every smooth manifold can be triangulated, is it true for orbifold? Is it a known result? If yes, is there any reference? reply to the comment : G does not need to be any subgroup of Sn , any ...
1 vote
1 answer
134 views

Chain of interior of closed set

It is well known that a topological space with asending chain condition for open subsets is called Noetherian. Is there any characterizations or a nice property for a Hausdorff topological space such ...
1 vote
0 answers
152 views

Stone cech compactification of a zero dimensional topological space

Let $X $ be a zero dimensional topological space, that is, a topological space with a basis of clopen sets. Is there any characterization for the ston cech compactification for such a space?
2 votes
1 answer
654 views

Triangulation induces regular CW complex structure

If a topological set is triangulable, dose the triangulation map gives it the (regular) CW complex structure? From definitions, I see it seems to be, but I am not that sure, for may exist some strange ...
3 votes
1 answer
267 views

In what sense is every element of $H_2(G)$ "represented by a free action on some surface"

(This is a cross-post of this unanswered math.stackexchange question) In Edmond's 1982 paper Surface Symmetry II, at the bottom of page 145, he writes: "Corollary - If $G$ is a split nonabelian ...
10 votes
2 answers
452 views

Quotient of $S^3$ by Montgomery and Zippin's "wild involution"

In 1952, Bing showed the existence of a topological involution of $S^3$ with fixed point set the Alexander horned sphere, demonstrating that $S^3$ has finite-order homeomorphisms not conjugate to ...
12 votes
4 answers
1k views

Elementary proof that knot complements are path-connected

The complement of any (topological) knot is path-connected. More precisely, if $K$ is a subset of $\mathbb{R}^3$ (or $S^3$) homeomorphic to $S^1$, then $\mathbb{R}^3\setminus K$ (or $S^3\setminus K$) ...
2 votes
0 answers
109 views

Compare two topologies: Three 2-tori inside $S^3 \times S^1 \# S^2 \times S^2$ glued from two different diffeomorphisms

We like to ask for the comparison of two topologies of three 2-tori inside the same 4-manifolds glued from two different diffeomorphisms (see the end). Given an embedded torus $T$ with trivial normal ...
3 votes
2 answers
181 views

A conjecture on antipodes and Jordan curves on the sphere

I got the following conjecture: Let $C$ be a Jordan curve on the sphere $S:=S^2$ and let $A$ and $B$ be the connected compnents of $S-C$. Then there is a pair of antipodes $a$ and $b$ such that $a\in ...
5 votes
1 answer
320 views

Ramified covers of S^n

This question has been inspired by covering 3-torus post. Is it true that any good (smooth, compact, oriented) $n$-manifold can be mapped to $S^n$ in such a way that the map is true covering away ...
2 votes
1 answer
166 views

What is the most symmetric configuration of four 2-surfaces linked in $S^4$?

What are some of the most symmetric configurations of four 2-surfaces linked in the 4-dimensional sphere $S^4$? To make a lower-dimensional analogy, recall that in 3-dimensional sphere $S^3$, we can ...
3 votes
0 answers
359 views

Cubical approximation theorem for cubical complexes

A version of the simplicial approximation theorem states that a continuous map between finite simplicial complexes is homotopic to a simplicial map after subdividing the domain. I have found a claim ...
3 votes
0 answers
104 views

A link of four 2-tori $T^2$ in $S^2 \times S^2$

Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary with their three $S^1$ boundaries of $T^3$ cyclic permuted to obtain a new 4-...
3 votes
0 answers
106 views

A link of four 2-tori $T^2$ in $S^3 \times S^1 \# S^2 \times S^2 \# S^2 \times S^2$

Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary to obtain a new 4-manifold: $$(S^4 \smallsetminus D^2\times T^2) \cup (S^4 \...
0 votes
0 answers
850 views

Meaning of Regular Neighborhood for Homology Basis Curves in $S_{g,2}$

I have been trying to understand the meaning of the expression "regular neighborhood" in the context described below, but I'm stuck: We have a collection of curves $c_i$ for $i=1,2,..,n$ embedded in ...
4 votes
2 answers
619 views

Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...
10 votes
1 answer
1k views

Open (resp., closed) balls homeomorphic to open (resp., closed) discs on the plane

Let $\Sigma$ be a compact (smooth) surface, with a geodesic metric $d$ (compatible with the topology of $\Sigma$). Let $x \in \Sigma$, and suppose you have the following: for every $r<1$, the open ...
8 votes
2 answers
689 views

Topological characterization of injective metric spaces

Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ \...
23 votes
1 answer
2k views

Is the normal bundle of a torus trivial?

Question: Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial? What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the $...
16 votes
10 answers
6k views

Undergraduate Topology

I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover. At my institution, real analysis is not a prerequisite for the course, so it is more than ...
4 votes
0 answers
352 views

A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$ where $a,b$ are two given points in the plane and $\lambda$ is a constant. Now we consider the ...
11 votes
1 answer
1k views

Simply connected noncompact surfaces

Is there a theorem saying that any noncompact, simply connected topological surface is homeomorphic to the plane ? There seems to be many well-known results about the classification of compact ...
-3 votes
1 answer
230 views

Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]

Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)? If so, please show me how to construct it.
13 votes
0 answers
515 views

pizza lemma (topology)

given six real-analytic arcs in the unit disk $D$, each of which connects the origin to a boundary point, and no two arcs meet anywhere except at the origin, and the arcs meet at equal (60 degree) ...
10 votes
1 answer
346 views

A forked plane continuum

I came up with this question while trying to solve the following MO one: Does every connected set that is not a line segment cross some dyadic square? Suppose $C$ is a plane continuum (i.e. a ...
3 votes
0 answers
93 views

When closed subsets have finitely many connected componenets

Let $X$ be topological space such that every its closed subset has finitely many connected componenets. Is there any charactrization for such topological space?
8 votes
6 answers
2k views

Uncountable preimage of every point

Let $f:[0,1]\to [0,1]$ be a continuous function. Must it have a point $x$ that $f^{-1}(x)$ is at most countable? Added: Must it have a point $x$ that $dim_H(f^{-1}(x))=0$ ? ($dim_H$ means the ...
4 votes
1 answer
177 views

Symmetry of a distance metric for a generating set of Topology

I was trying to prove that $\epsilon$-balls defined based on the shortest travel-time distance in a transportation network is a valid generating set for a topology of points on a transportation ...
2 votes
1 answer
131 views

Shrinkable decompositions with uncountably many non-degenerate elements?

Let $\mathcal D$ be an upper semicontinuous decomposition of $\mathbb S^n$ and let $\mathcal D'\subset\mathcal D$ be the set of non-singletons. The decomposition space $^{\mathbb S^n}/_{\mathcal D}$ ...
1 vote
0 answers
114 views

When Max(R) is Hausdorff space? [duplicate]

Let $R$ be reduce commutative ring with identity (a commutative ring such that $a^n$=0 ($a\in R$) implise $a=0$) and $Max(R)$ be the set of all maximal ideals of $R$. The hull-kernel (or Zariski ...
2 votes
0 answers
224 views

cross-sections of a sphere bundle

Let $M$ be a $m$-manifold and $M_0$ a submanifold of $M$. Let $X$ be a pointed topological space. In the paper On the homology of configuration spaces, Bodigheimer-Cohen-Taylor, Topology 1989, ...

1
3 4
5
6 7