The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key ...

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20
votes
2answers
931 views

The “Easiest” non-smoothable manifold

In 1960, Kervaire found the first example of a PL-manifold which does not admit a smooth structure. Since then, I understand that there are many examples of non-smoothable manifolds that can be built. ...
19
votes
1answer
473 views

Decomposition of a manifold

As a kind of aside to this question, where one of the answers assumed that if $S^n=X \times Y$ then we can assume that $X$ and $Y$ are manifolds. If we have a manifold $M$, such that $M$ is ...
8
votes
1answer
433 views

Equivalence of knots: ambient isotopy vs. homeomorphism

I am looking into knot theory and have found two different definitions stating that two knots $K_1$ and $K_2$ are equivalent, namely the concept of an ambient isotopy: These two knots are ambient ...
4
votes
1answer
160 views

cup product in cohomology ring of a suspension

Let $X$ be a CW-complex. Let $\Sigma$ be suspension. Let $R$ be a commutative ring. Is the cup product of $$ H^*(\Sigma X;R)$$ trivial? How to prove? Where can I find the result?
5
votes
1answer
925 views

The Join of Two Copies of $S^1$

So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I'm having trouble "seeing" this. I'd prefer something that appeals to geometric intuition, but more formal ...
10
votes
2answers
4k views

Understanding the Equation of a Möbius Strip

I am in HL Math and trying to finish my IA. My topic is the Möbius band. The only problem is, I do not understand the formula that defines it and everywhere I have looked has just given me a ...
7
votes
1answer
367 views

How to Classify $2$-Plane Bundles over $S^2$?

I'm curious how one can classify the bundles over a given manifold. I recently read this paper on classifying $2$-sphere bundles over compact surfaces. A lot of the concepts went over my head since ...
8
votes
1answer
122 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic)

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
6
votes
1answer
273 views

Question on “up to isotopy” when attaching two spaces

Let $M$, $N$, $A$, $B$ be topological spaces (or manifold) such that $A$ and $B$ are subspaces in $M$, $N$ respectively. Let $f: A \to B$ and $g:A \to B$ be homeomorphism and assume that $f$ and $g$ ...
4
votes
1answer
461 views

How does handle attachment work in Morse Theory

I am reading R.E.Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. I can't understand the 2nd-paragraph of p.101, where they explain framings on the attaching sphere. In particular I cannot ...
3
votes
1answer
505 views

A 3-manifold with fundamental group isomorphic to a surface group.

Let $M$ be a 3-manifold (the case I am interested is $M$ closed orientable connected hyperbolic); suppose $\pi_1 (M)$ is isomorphic to the fundamental group of a (closed orientable connected) surface ...
2
votes
2answers
126 views

the knot surgery - from a $6^3_2$ knot to a $3_1$ trefoil knot

It is intuitive that one can simply doing a cut-gluing surgery to make a $6^3_2$ to a $3_1$ trefoil knot: e.g. from to All one needs to do it to cut the three intersections at the angle of ...
6
votes
2answers
257 views

Maps of discs into surfaces

Let $f:D \to S $ be a continuous map from the closed unit disc in $R^2$ into a closed surface of genus $g \geq 1$ such that $f|_{\partial D}$ is an embedding (homeomorphism onto its image) with Image ...
1
vote
1answer
304 views

Immersing punctured torus

I am looking for a proof (as elementary as possible) of the fact that punctured n-dimensional torus admits an immersion to $\mathbb{R}^n$. The 2-dimensional case seems to be evident, but I haven't got ...
33
votes
6answers
704 views

Is the complement of countably many disjoint closed disks path connected?

Let $\{D_n\}_{n=1}^\infty$ be a family of pairwise disjoint closed disks in $\mathbb{R}^2$. Is the complement $$ \mathbb{R}^2 -\bigcup_{n=1}^\infty D_n $$ always path connected? Here ...
10
votes
1answer
412 views

Equivalence of Definitions of Principal $G$-bundle

I've finally gotten around to learning about principal $G$-bundles. In the literature, I've encountered (more than) four different definitions. Since I'm still a beginner, it's unclear to me whether ...
11
votes
1answer
394 views

Space of homeomorphisms Homeo$(S^1)$ of $S^1$ deformation retracts onto $O(2)$

How can we prove that the space of homeomorphisms Homeo$(S^1)$ of $S^1$ (strong) deformation retracts onto the orthogonal group $O(2)$? I know that this result is proved by Hellmuth Kneser in his ...
9
votes
1answer
161 views

General relationship between braid groups and mapping class groups

I just finished correcting my answer on visualizing braid groups as fundamental groups of configuration spaces, and in the process became interested in the other pictorial definition of the braid ...
3
votes
1answer
80 views

A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph

The stable manifold theorem tell us: A local stable manifold $W^{s}_{loc}(x^{*})$, which is a differentiable manifold of class $\mathcal{C}^{r}$ and dimension $n_{-}, $ tangent to the ...
12
votes
1answer
676 views

Gluing a solid torus to a solid torus with annulus inside.

I was thinking the fact that if two genus $1$ handlebodies (solid tori) are glued via an orientation preserving homeomorphism of boundaries, the resulting manifold depends only on (up to isotopy) ...
9
votes
2answers
591 views

Does the Euler characteristic of a manifold depend upon the field of coefficients?

Define the Euler characteristic of a space $X$ to be $$\chi(X)= \sum_i \dim H_i(X, \mathbb Q)$$ This is obviously not necessarily well-defined for an arbitrary space $X$, so let $X$ be a manifold ...
7
votes
2answers
626 views

Which mapping tori are Seifert manifolds?

According to Orlik's lecture notes on Seifert manifolds (and the Wikipedia page on Seifert fiber spaces), a mapping torus over a 2-torus is a Seifert manifold if and only if it is the mapping torus of ...
5
votes
1answer
124 views

Embedding manifolds of constant curvature in manifolds of other curvatures

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space ...
2
votes
0answers
101 views

Surgery to unlink $S^1$ and $S^2$ in $S^4$ [closed]

Let us start with a $S^1$ and a $S^2$ are linked in $S^4$. Can I unlink the $S^1$ and $S^2$ by doing some surgery (with certain constraints described below, and let us say both $S^1$ and $S^2$ ...
2
votes
0answers
65 views

Embed $S^{p} \times S^q$ in $S^d$?

Can we embed $S^{p} \times S^q$ in $S^d$ with all the nice properties, what are the allowed values of $p$ and $q$ for $d=2,3,4$ where $p+q \leq d$? =For $d=2$= I suppose that we cannot embed $S^{1} ...
6
votes
2answers
222 views

Homotopically trivial $2$-sphere on $3$-manifold

Let $S^2$ be an embedded sphere in a $3$-manifold $M^3$ such that $[S^2]$ is trivial in $\pi_2(M)$. Can we find an embedded disk $D^3$ in $M$ such that $\partial D^3=S^2$?
4
votes
1answer
45 views

identify the topological type obtained by gluing sides of the hexagon

Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$ I do not understand how the ...
3
votes
1answer
149 views

Existence of Closed Curves around Bounded Components

I am stuck on part of a complex analysis proof that I think needs more justification than given. It's pretty purely a topological statement, but it may be that complex-analytic techniques would be ...
3
votes
0answers
182 views

Is such an infinite dimensional metric space, weakly contractible?

We counteract this answer by adding the rigidity assumption: Is there still a counterexample? Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: ...
3
votes
2answers
287 views

line equidistant from two sets in the plane

Suppose $A$ and $B$ are disjoint subsets of the plane, both closed, nonempty, and connected. Define $E(A, B)$ as the set of points in the plane equidistant from $A$ and $B$. For example, if $A$ is a ...
3
votes
3answers
749 views

Orientation reversing diffeomorphism

Why each sphere admits an orientation reversing diffeomorphism onto itself? (For even dimensional ones can we take conjugation map?) And why complex projective spaces do not admit? Is there a ...
2
votes
1answer
37 views

Two ambient isotopic curve segments, one has the length and the other does not

Let me ask if the following is possible: Let $L_1$ be some curve segment in the $\mathbb{R}^3$ space which has the length $1$. Let $L_2$ be some curve segment in the $\mathbb{R}^3$ space which you ...
2
votes
0answers
245 views

Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often ...
2
votes
2answers
220 views

intersection of decreasing path-connected spaces

If we have path-connected spaces $A_0 \supseteq A_1 \supseteq A_2 \supseteq \ldots$, is $\bigcap^\infty A_i$ path-connected? I was thinking that if we take $A_i$ to be a $1/i$-neighborhood of the ...
1
vote
1answer
211 views

How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like ...
5
votes
1answer
64 views

Minimal-dimension example of (open) subset of $\mathbb{R}^n$ with trivial first cohomology but nontrivial fundamental group

As a follow-up to this question, I was wondering what dimension provides the minimal counterexample to the claims: If $U\subseteq\mathbb{R}^n$ is an open connected set with trivial $H^1(U)$, then ...
5
votes
3answers
169 views

when is the region bounded by a Jordan curve “skinny”?

How can I formalize and prove the following intuition?: Picture a very skinny rectangle, one with base length 1 and sides length $\epsilon$. Or imagine a very flattened ellipse. The interiors of ...
4
votes
1answer
99 views

Degree of maps and coverings

Following a recent question I had concerning degree $1$ maps from spheres, I came up with an assumption, which might either be very easily proven false, or, if not, still hasn't been answered. It goes ...
4
votes
2answers
203 views

What is smoothness needed for?

We can either define a knot to be (1) a smooth embedding $S^1 \hookrightarrow \mathbb R^3$ or (2) a piecewise linear, simple closed curve in $\mathbb R^3$ Then these two definitions are ...
3
votes
1answer
53 views

Homeomorphisms of different spheres

It is pretty straightforward to show that if $X$ is a Banach space under two equivalent norms, then the respective open unit balls $B_1$ and $B_2$ are homeomorphic only by showing that $B_i$ is ...
3
votes
1answer
192 views

Topological shape of sphere-like equations with complex radius

For fixed complex number $s≠0$, what 4-dim shape is given by complex solutions $z,w$ of $z^2+w^2=s^2$ and higher dimensional version 2N-dim shapes of $z_1^2+z_2^2+...+z_N^2=s^2$ ?
3
votes
1answer
569 views

the restriction of a homeomorphism on a subset

Let $X$ be a topological space and $f:X\to X$ be a homeomorphism. Then the induced map $f_1:\pi_1(X,x)\to\pi_1(X,fx)(\cong \pi_1(X,x))$ is an isomorphism (automorphism up to conjugate). In the ...
1
vote
0answers
37 views

Does this container exist?

EDIT: Note that the object I'm seeking needn't have anything to do with water or actual containers; those are just used to convey the idea. I'm trying to find a container that, when turned with ...
1
vote
1answer
55 views

Geometric Intuition for “Right-Veering” Property of $f$ in MCG(S)?

let $S$ be a compact surface with non-empty boundary, let $\alpha : [0,1] \rightarrow S$ be a Properly-embedded arc (meaning both endpoints of the arc are in $\partial S$) and let $f$ be an element ...
1
vote
1answer
106 views

Plane models from the “word”

I have a "word" for a plane model $abacdc^{-1}db^{-1}$. From what I reckon, it's a torus. But I am not too sure of it. I sketched it up and did some "adjustments". Could it be a projective plane ...
1
vote
1answer
77 views

Number of connected components of the complement of a closed curve.

Let $\gamma:[0,1] \rightarrow \mathbb{R}^2$ be a continuous, closed curve (i.e. $\gamma(0) = \gamma(1)$). My question is about the number of connected components of the complement $\mathbb{R}^2 ...
1
vote
1answer
356 views

Pinched torus generalization

The pinched torus is homeomorphic to a sphere with two (different) points identified.           What is the name and topological structure of the ...
1
vote
2answers
282 views

When does an embedded $2$-torus bound a solid torus in $3$-manifolds?

This is a simple version of the question asked here. Let $M$ be a compact 3-manifold without boundary, $\mathbb{T}^2$ be the standard 2-torus, and $i:\mathbb{T}^2\to M$ be an embedding. (*) Assume ...
0
votes
0answers
59 views

Is this octogon topologically equivalent to the Klein Bottle?

Note: this is an extension of a previous problem (identify the topological type obtained by gluing sides of the hexagon ) where a hexagon was considered. Is the space below also a Klein bottle ...
0
votes
1answer
49 views

Homeomorphism between $\Bbb S$\ $\lbrace-1\rbrace$ and $\left(-1,1\right)$

I Know that $\Bbb S$\ $\lbrace-1\rbrace=\lbrace e^{i\theta}:\theta\in \left(-\pi,\pi\right)\rbrace=\lbrace e^{i\pi t}: t\in\left(-1,1\right)\rbrace$ Let $f:\Bbb S$ \ ...