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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3answers
31 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
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0answers
18 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
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0answers
67 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 ...
-1
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0answers
37 views

Topological Equivalence of Metric Spaces [closed]

Suppose we have two different metric spaces $(X,\phi)$ and $(Y,\psi)$. I need to show that the metrics $\phi$ and $\psi$ are equivalent metrics. Using a sterographic projection, I've shown that if we ...
0
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0answers
30 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
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1answer
61 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 ...
0
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0answers
43 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
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0answers
63 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
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1answer
51 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 ...
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0answers
35 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
35 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
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1answer
47 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
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0answers
33 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?
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0answers
28 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
508 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
111 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
79 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
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0answers
35 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
61 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
37 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
193 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
207 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 ...
3
votes
2answers
165 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
44 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
58 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 $\mathbb{R}^3$?
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1answer
10 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
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0answers
48 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
179 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
118 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
56 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
70 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
137 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 ...
1
vote
1answer
41 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
44 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 ...
1
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0answers
55 views

The number of Connected Components of a topological space

Let $X$ be a topological space, and $Y$ a closed subset of $X$. If we can express $Y$ as a finite disjoint union of connected closed subsets of $X$, is this expression unique or at least the number of ...
0
votes
1answer
89 views

Finite topological space

At http://math.stackexchange.com/questions/1528995/finite-topological-space the user asked "Let $X$ be a Hausdorff topological space such that every closed subset has finitely many connected ...
0
votes
1answer
46 views

Is there a direct proof that an $(n-1)$-simplex in a subdivision of the standard $n$-simplex is a face of at most two of its $n$-simplices?

There are many books and articles that prove Sperner's Lemma. Almost all that I have looked up happily take the following as obvious. If $\mathcal{S}$ is a simplicial subdivision of the standard ...
3
votes
1answer
58 views

$c_1^3$ of 6 manifolds

For a closed oriented smooth 4 manifold $X$ we have $c_1^2(X) := 2e(X) + 3σ(X)$, $c_1$ is the first Chern class and $σ$ is the signature. For 6 manifolds is there such a relation with $c_1^3$? I'd ...
4
votes
1answer
61 views

When are mapping tori isomorphic as bundles over the circle?

Suppose $\Sigma$ is an orientable genus-$g$ surface (possibly with boundary). The mapping torus corresponding to an orientation-preserving diffeomorphism $\phi: \Sigma \to \Sigma$ is the quotient ...
1
vote
1answer
55 views

Construction of Grassmann manifolds

Is there a way to construct the Grassmann manifold via block matrices? For example the upper triangular matrices stabilize the (coordinate) basis of $\mathbb R^n$.
8
votes
0answers
132 views

Prerequisites for studying Perelman's proof of the Geometrization Conjecture

I want to set a course toward understanding Perelman's proof of the Geometrization Conjecture. I realize this will be a lengthy undertaking, but hopefully only on the order of one to two years. I am ...
2
votes
1answer
56 views

The Klein bottle is homeomorphic to the boundary of the product of the Möbius band with a disk

Can someone please give me a hint or the intuition in how to prove that the Klein bottle $\cong \partial$(Möbius strip $\times D^1 $ ) where $\cong$ means homeomorphic.
3
votes
0answers
48 views

Simplicial homology for infinite complexes

Simplicial homology can be viewed as a covariant functor from the category of finite simplicial complexes with continuous maps over support polyhedra, to the category of sequences of abelian groups. A ...
0
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0answers
18 views

Can plane be tiled with polygons

I'm working on cellular automaton, where each cell has K number of friends. For example cellular automaton with 8 friends and particular rule is Conway's Game of Life. I want to draw grid for any ...
1
vote
2answers
53 views

triviality of vector bundles with the reduced homology of base space entirely torsion

Let $\xi$ be an $n$-dimensional vector bundle over a manifold $M$ such that the reduced cohomology $\tilde H^*(M;\mathbb{Z})$ is entirely torsion (every element has finite order under addition). ...
3
votes
1answer
38 views

Betti number of nonnegative Ricci curvature and positive scalar curvature closed 3-mfd

Suppose that $M^3$ is a closed 3-manifold with nonnegative Ricci curvature and positive scalar curvature, I think $b_1(M^3)\leq 1$. Is this right and is there a quick cut proof?
2
votes
1answer
32 views

Universal cover of boundary

Let $M$ be a compact manifold-with-boundary and $B$ a component of $\partial M$. Let $\tilde{M}$ be the univeral cover of $M$ with infinite-sheeted covering map $p:\tilde{M} \to M$. I wonder about the ...
1
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0answers
11 views

A surface is essential iff each component of it is essential?

First, for the terms, A surface is essential if it is both incompressible and boundary-incompressible. I want to show that A surface S is essential if ans only if each components of S is essential. ...
1
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0answers
47 views

Classification of Torus bundle over $\mathbb{S}^1$ [closed]

$T^2$ has a fixed free isometry group as $D_8$, is this true that $T^2$ bundle over $S^1$ have three type?
2
votes
0answers
44 views

Classification of closed flat three dimensional manifold? [closed]

Obviously $T^3$ is one type. $K^2\times S^1$ is also one type. Maybe $T^2\tilde{\times}\mathbb{S}^1$ bundle is also one type.