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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3
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0answers
37 views

Gluing two solid tori by a homeomorphism of their boundarries.

I am aware that we get all lens spaces by gluing two solid tori by their boundaries. My question is, do we get more spaces besides lens spaces? in other words, do all homeomorphisms of the boundaries ...
0
votes
0answers
27 views

The condition to ba an ambient isotopy

Let me ask the following question about ambient isotopies. https://en.wikipedia.org/wiki/Ambient_isotopy An article in Wikipedia says: A continuous map F:M * [0,1] -> M is defined to ...
0
votes
0answers
21 views

Diffeomorphisms with fixed subsets and isotopies

I asked a much more general version of this question previously, but I realize now that it was so broad that there probably are no good general answers, so I want to make a more specific question. ...
1
vote
1answer
26 views

Isotopies with fixed subsets

Let $M$ be a smooth manifold. Let $f:M\rightarrow M$ be a diffeomorphism which is smoothly isotopic to the identity. Let $X\subset M$ be a compact subset such that $f|_X = id_X$. Under what ...
7
votes
1answer
52 views

Reference on manifolds with corners

Is there a systematic treatment of (finite dimensional) manifolds with corners in the literature which carefully introduces all usual differential topological notions (submanifolds, embeddings, etc.) ...
0
votes
1answer
51 views

Estimate on Size of Convex Hull

Let $X$ be a bounded subset of $\mathbb R^3$, and denote by $Conv(X)$ the convex hull of $X$, i.e, the smallest convex subset of $\mathbb R^3$ containing $X$. As $X \subset Conv(X)$ by definiton, the ...
2
votes
1answer
41 views

Smooth vector field vanishing at exactly 6 points

Let $ E $ be the ellipsoid in $ \mathbb{R}^3 $ given by, $$ E = \left \{ (x,y,z) \mid \frac{x^2}{9} + \frac{y^2}{4} + z^2 = 1 \right \} $$ Find a smooth vector field $ H : \mathbb{R}^3 \rightarrow ...
3
votes
1answer
41 views

Extension of Smooth Functions on Embedded Submanifolds

In Lee Smooth Manifolds, this problem is given: if $S \subset M$ is smoothly embedded and every $f \in \mathcal{C}^{\infty}(S)$ extends to a smooth functional on $\textit{all}$ of $M$, then $S$ is ...
0
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0answers
29 views

Fundamental group from triangulation

How do I calculate the fundamental group of surface if I have been given it in triangulated form. I have attached an example triangulation I think I may have to simplify it further to a tree and ...
0
votes
1answer
21 views

Show that chordal metric is topologically equivalent to the Euclidean metric

Consider $$d(x,y)=\frac{2\|x-y\|}{(1+\|x\|^2)^{1/2}(1+\|y\|^2)^{1/2}},\hspace{5mm}x,y\in \mathbb{R}^n.$$ $d$ is a metric in $\mathbb{R}^n$ known as chordal metric. I want to show that this metric is ...
10
votes
2answers
528 views

Why are invariants of knots and manifolds important or useful?

It is easy to define invariants that completely classify knots; however, this is computationally infeasible, so is it computationally efficient invariants that are important? Why? Do mathematicians ...
0
votes
0answers
37 views

Relation between Alexander duality and linking numbers

I just have the feeling that there must be some relation between Alexander duality and linking numbers, but I don't know what is that. Will anyone tell me anything about that? Or could anyone give ...
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 ...
0
votes
1answer
45 views

Knotted cross-sections of unknotted spheres

Livingston's notes on concordance mention "embeddings of $S^2$ into $\mathbb{R}^4$ which are unknotted, but have non-trivial knots as cross-sections. There are other such unknotted two spheres with ...
1
vote
1answer
120 views

Proof of the bigon criterion

In Farb and Margalit's A Primer on Mapping Class Groups we have the following Proposition 1.7: Two transverse simple closed curves in a surface $S$ are in minimal position iff they do not form a ...
0
votes
0answers
33 views

universal coefficient theorem for mod p cohomology

In the book Algebraic Topology, Allen Hatcher, p. 266, Corollary 3A.6 (b): Question: I want to rewrite the above statement into a cohomology version. If I replace all homologies with cohomologies, ...
1
vote
1answer
76 views

Connected sum of two “same” Klein bottles

If I take sphere and remove two open disks from it and on the boundary of that space I make identification like on the picture, what do I get? Are both of those objects Klein's bottles? This is what ...
1
vote
3answers
39 views

What is the distance of two circles put on eachother?

If one takes two circles (lets say on a straight cilinder), and bring the circles closer and closer to eachother. Will the distance between them goes to zero, or can you say the distance is for ...
1
vote
1answer
42 views

Unbounded, closed, star-shaped set contains ray

I am trying to prove the following statement: Let $R$ be a real closed field (such as the real numbers). Let $M\subseteq R^n$ be a semi-algebraic set, i.e. a set which is defined by a Boolean ...
0
votes
0answers
76 views

How to prove, no tame knot is isotopic to a non tame knot?

Please let me ask the following question. I have read in Wikipedia, quote: A polygonal knot is a knot whose image in R^3 is the union of a finite set of line segments. A tame knot is any knot ...
0
votes
0answers
32 views

A. A. Markov's paper on insolubility of the homeorphy problem [duplicate]

I am aware that this has been asked before, but the paper is nowhere to be found online, the provided link in the old thread leads to nowhere, and I'm really at wits end to find this paper, can anyone ...
1
vote
1answer
91 views

Self-indexing Morse function on a torus which is a height function

A Morse function $f: \Bbb T^2\to [0,2]$ is called self-indexing if $f^{-1}(n)$ is the set of critical points of index $n$. It is relatively easy to see that on any compact manifold, any Morse function ...
1
vote
0answers
53 views

Compact 1-manifolds

I want to find all compact 1-manifolds. I think these will be lines, with fixed endpoints, and also the case where the endpoints meet. Hence I think all compact 1-manifolds are homeomorphic to the ...
0
votes
0answers
75 views

Identification space of square. Net, triangulation and surface classification

Space Z is made as an identification space of unit square $Q=${$(x,y) | 0\leq x, y \leq 1$} by making the following identifications: $ (0,y)$~$(1,y) $ for all $0\leq y\leq 1 $, $ ...
2
votes
1answer
60 views

Regular neighbourhoods of non-orintable surfaces in $S^4$

Suppose that $F \subset S^4$ is a non-orientable surface. Let $N \subset S^4$ be a regular neighbourhood of $F$. Clearly, the boundary of $N$ is a circle bundle over $F$. Which is its Euler number? My ...
1
vote
0answers
40 views

Simply connected compact subsets of $\mathbb R^2$

Is every simply connected compact subset of $\mathbb R^2$ weakly contractible, i.e. all homotopy groups vanish?
1
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0answers
39 views

Plane curves admitting several ways to seal it up by the disc

Let $\gamma:S^1\to\mathbb R^2$ be an immersed closed curve with only isolated transversal self-intersections. Say that $\gamma$ is sealed by the disc if there is an immersion $f:D^2\to\mathbb R^2$ ...
5
votes
1answer
52 views

Components of the space of immersions 2-manifold into $\mathbb R^3$

Let $M$ be a $2$-sphere with $g$ handles. Consider the space of maps $M\to \mathbb R^3$, which are immersions [i.e. smooth maps with nondegenerate differential in each point $x\in M$], with ...
1
vote
0answers
44 views

What is the fundamental group of a cone?

I am reading an article on orbifolds and it describes the cone as the quotient of unit 2-dim. disc by a finite cyclic group of rotations. But how is it's fundamental group finite cyclic?
1
vote
0answers
35 views

Reducible Heegaard splittings can be written as connected sums

Recall that a Heegaard splitting is a decomposition of a three manifold into a triple $(V,W,S)$ where V and W are solid handlebodies meeting along their common boundary S. A Heegaard splitting is ...
8
votes
2answers
531 views

Length of the main diagonal of an n-dimensional cube

Find the length of a main diagonal of an n-dimensional cube, for example the one from $(0,0,...,0)$ to $(R,R,...,R)$ I tried to use induction to prove that its $\sqrt{n}R$ but I'm stuck on writing ...
6
votes
1answer
121 views

Is $(\#^k \Bbb{RP}^2) \times I$ an $\mathbb{RP}^2$-irreducible 3-manifold?

Consider $S$ a surface homeomorphic to a connected sum of $n$ projective planes, $n \geq 2$. Can there be a two sided projective plane embedded in $[-\epsilon,\epsilon]\times S$?
2
votes
1answer
103 views

Maps from Sum of Projective Planes to Circle

this is a problem from Lee's Topological Manifolds, 11-21. It asks the following: What compact, connected surfaces $M$ admit a non-nullhomotopic map to the circle? So, we use the classification of ...
2
votes
0answers
36 views

Platonic Hopf. Given the vertices of a tetrahedron circumscribed by unit sphere, find the stereographic projection of inverse Hopf fibers

I am trying to find the equations in $\mathbb{R}^3$ for the fibers of the four vertices of the tetrahedron circumscribed by the unit sphere. I want to find $s\circ h^{-1}$, where $s$ is the ...
2
votes
1answer
68 views

what does Whitney sum of vector bundles correspond to in the k-theory KO?

Let $X$ be a $CW$-complex and $\text{Vect}^n(X)$ the collection of $n$-dimensional real vector bundles over $X$. Let $$ \text{Vect}^*(X)=\bigoplus_{n=0}^\infty \text{Vect}^n(X) $$ with addition ...
6
votes
1answer
39 views

Visualizing order 3 mapping class of genus 2 surface

Let $\Sigma_2$ be a closed genus $2$ surface. There exists an orientation-preserving diffeomorphism $f:\Sigma_2 \rightarrow \Sigma_2$ of order $3$. The diffeomorphism has $4$ fixed points (each, of ...
10
votes
2answers
239 views

Which groups act freely on $S^n$?

When $n$ is even, it is easy to classify groups which act freely on $S^n$ using degree theory: if $G$ acts on $S^n$, then associating to each element $g \in G$ the degree of the map obtained from ...
17
votes
2answers
268 views

What closed 3-manifolds have fundamental group $\Bbb Z$?

For certain small groups, it is easy (and desirable) to classify closed (and orientable if necessary) 3-manifolds with that group as their fundamental group. (Essentially due to Waldhausen is that for ...
4
votes
2answers
177 views

group actions on spheres

Let $\mathbb{Z}/2$ act on the $m$-sphere $S^m$ freely and properly discontinuously. If the action is not trivial, can we conclude that the action is homotopy equivalent to the antipodal action? That ...
3
votes
0answers
52 views

commutivity of suspension and quotient space

Let $(A,A_0)$ be a $CW$-pair such that $A_0$ is a $CW$-subcomplex of $A$. Let $\Sigma$ be the suspension. Is $\Sigma(A/A_0)$ homotopy equivalent to $\Sigma A/\Sigma A_0$ or not?
3
votes
1answer
61 views

Construction of an immersion of $T^3$ − point in $\mathbb{R}^3$?

Let $p\in T^3$. How do I construct an immersion of $T^3\setminus\{p\}$ in $\mathbb{R}^3$?
-1
votes
1answer
11 views

what are the symmetries and flags of tetrahedron?

I know that |rot(tetrahedron) | = 12 ( i know how we came up with this number ) my question what is the number of symmetries in tetrahedron ? is it 12 or 24? if is it 24 can anyone explain to me how ...
2
votes
0answers
60 views

Difference between algebraic topology and geometric topology [closed]

What are the main differences between these two areas? Does geometric topology in general, use more analytic techniques? Which one would most consider harder? Is one more general than the other?
5
votes
1answer
185 views

Non-diffeomorphic structures on the sphere

How many smooth structures are there on $S^2$, $S^3$, and $S^4$ up to diffeomorphism? I looked around and couldn't find an answer; two books I have say different things on the subject. I know one of ...
7
votes
1answer
123 views

Does it follow that $Y$ is homotopy equivalent to $S^{n-1}$?

Let $M$ be a compact connected $n$-manifold (without boundary), where $n \ge 2$. Suppose that $M$ is homotopy equivalent to $\Sigma Y$ for some connected based space $Y$. Does it follow that $Y$ is ...
1
vote
1answer
97 views

Collar neighborhood theorem- Cobordism

I am studying cobordism theory and I basically follow Milnor's book, Characteristic classes. I want to prove that cobordism is a transitive relation so I need the collar neighborhood theorem. In ...
3
votes
0answers
73 views

When can a polygon with some edge identifications be embedded in $S^3$?

Let $P$ be a polygon, and therefore a topological disk. Suppose we make some identifications on its edges, possibly identifying 2 or more edges of the polygon to a single edge, to get a 2-complex $K$ ...
3
votes
1answer
148 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 ...
2
votes
1answer
47 views

embeddings of projective spaces into Euclidean spaces

Let $\mathbb{R}P^n$, $\mathbb{C}P^n$, $\mathbb{H}P^n$ be the real, complex, quaternionic projective spaces resp. I want to find all $n$ such that $\mathbb{R}P^n$ can be embedded into ...
7
votes
1answer
50 views

Is the union of an increasing sequence of topological copies of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?

Let $M$ be an $n$-dimensional topological manifold, and let $(U_k)_{k \in \mathbb{N}}$ be an increasing sequence of open sets $U_k \subset M$ such that for each $k \in \mathbb{N}$, $U_k$ is ...