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 ...

learn more… | top users | synonyms (1)

5
votes
1answer
186 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
124 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
111 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
74 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
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 ...
2
votes
1answer
50 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 ...
1
vote
0answers
74 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
61 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
61 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
85 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
75 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 $\mathbb R^n$.
9
votes
0answers
161 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
71 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
52 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 ...
1
vote
2answers
67 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
41 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?
3
votes
1answer
52 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
vote
0answers
12 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. ...
2
votes
0answers
55 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.
3
votes
1answer
64 views

Explain unoriented $S^2$ bundle over $S^1$.

In Hamilton's classification of closed 3-d nonnegative Ricci curvature manifold, unoriented $S^2$ bundle over $S^1$ is one of the possible type. Who can give me a description of it? Many thanks!
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 ...
1
vote
0answers
48 views

Finiteness of Lusternik-Shnirelman category

Are there conditions on a topological space $X$ under which its Lusternik-Shnirelman category is countable (or even finite)? "Countable Lusternik-Shnirelman category" means that $X$ can be covered by ...
0
votes
0answers
26 views

Labeling the (p,q,r)-pretzel knot with transpositions from S4

For what values of p,q, and r, can the (p,q,r)-pretzel knot be labeled with transpositions from $S_4$? I'm kind of stuck on how to approach this one. All I've got so far is that there are six ...
0
votes
1answer
64 views

punctured real projective space

Let $\mathbb{R}P^m$ be the real projective space and $X=\mathbb{R}P^m\setminus \{*\}$ be the punctured space by removing one point. How to get the cohomology ring of $X$ with integer coefficient? Is ...
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 $\partial D^3=S^2$?
1
vote
0answers
41 views

Definition of One-point compactification

My question is an elementary one but I can't seem to find the formal definition. If $M$ and $N$ are topological spaces and $\dot{M}$ is the one-point compactification of $M$ then how is a map $f:M ...
3
votes
1answer
131 views

Are PL embeddings homotopic to smooth ones?

In particular, I'm interested in the case we have a PL embedding $f : S^1 \to M$ (for a smooth manifold $M$): can it be homotoped to a smooth embedding? I'm not very familiar with PL stuff, so I may ...
2
votes
0answers
44 views

Probabilistic proof for sphere covering upper bound

I would like to show an upper bound for the number of $d$-dimensional spheres needed to cover some closed, bounded subset of $\mathbb{R}^d$, like a cube or another sphere. I could do this by placing ...
6
votes
2answers
88 views

Generalized Jordan-Brower separation theorem

Suppose $M^{n+1}$ is a closed connected smooth manifold and $N^n$ is a closed connected smooth embedded submanifold of $M^{n+1}$. What's the weakest condition under which the Jordan-Brower ...
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 ...
0
votes
0answers
17 views

Fiber monodromy of complex multiplication

Consider the map $f: \mathbb{C}^2 \to \mathbb{C}$, with $f(z,w) = zw$. I read a remark that if we take a circle around the origin (say the usual loop starting and ending at 1) and we lift it to a map ...
4
votes
1answer
59 views

Does every torus $T\subset S^3$ bounds a solid tours $S^1\times D^2\subset S^3$?

I want to show this by using Alexander's Theorem's proof method. So here's what I thought. As I surger $T$, I have 2 $S^2$. So one bounds $S^1$ and the other bounds $D^2$. By reversing the surgery, ...
0
votes
0answers
34 views

Classification of symmetric space of non compact type

Is there a classification of rank one symmetric space of non compact type ? Remark, for rank 1, There are three families : 1) hyperbolic n-space, corresponding to the Lie group SO(n,1). 2) complex ...
1
vote
1answer
22 views

Is a continuous family of contractible spaces simultaneously contractible?

Let's say I have a surjective, continuous map $f: X \to Y$, and there is a deformation retract of the fibers $f^{-1}(y)$ to a point for every $y \in Y$. Is it always the case that there is a ...
0
votes
1answer
43 views

$\mathbb{Z}^2$ acts on $\mathbb{R}^2$ by translation is 'separable'

I have to study the following G-Action $$ \begin{cases}\mathbb{Z}^2 \times \mathbb{R}^2 & \longrightarrow \mathbb{R}^2 \\ (m,n)\times (x,y) & \longmapsto (x+m,y+n) \end{cases} $$ That is, as ...
0
votes
1answer
32 views

Jordan curves in compact subset in $\mathbb R^3$ [closed]

Always exist a curve in compact and convex subset of $\mathbb{R}^3$?
1
vote
0answers
61 views

Surgery presentation for an abstract open book decomposition

Suppose $(\Sigma,\phi)$ is an abstract open book whose monodromy is expressed as a product of Dehn twists about boundary-parallel curves. Is there a standard way to produce a surgery presentation of ...
3
votes
1answer
40 views

Topological embeddings of a $n$-ball $B^n$ in $\Re^n$: is the image always an open of $\Re^n$?

I've an elementary doubt about topology and I just can't find the answer (nor I'm able to have an opinion): is it true or false that if $S$ is a subspace of $\Re^n$ homeomorphic to the open ball ...
0
votes
0answers
31 views

Intuition of transversality equation

I am studying Differential topology from Guillemin / Pollack and unfortunately i cannot understand ıntuition of Transversality equation Let $f$ be a smooth map between smooth manifolds $X$ and $Y$ ...
2
votes
2answers
98 views

Prove there is a line that cuts the area in half

Suppose you are given two compact, convex sets A and B in the plane, prove there exists a line such that the area of A and B is simultaneously divided in half. Can you help me with this proof? What I ...
2
votes
1answer
56 views

Cell-decomposition

My professor said that this (picture) is an example for a non-cell-decomposition because there is missing an vertex at the end of the edge. He also claimed that we have $2$-faces, $3$-edges and ...
1
vote
1answer
90 views

Calculating Berry phase for parallel transport of vector around closed path on sphere.

I am following the text here which explains that we are transporting a vector $\mathbf{v}$ around a closed circular loop (C) on a sphere under the constraints $\mathbf{e}_3\cdot\mathbf{v}=0$ and ...
0
votes
1answer
44 views

Space as a network. Is it possible to model topological properties of Euclidean (or non-Euclidean) space using a discrete network of points?

I've read Wolfram's book 'A New Kind of Science', which was very interesting and introduced me to the topic of cellular automata. But he also introduces the idea to model the physical space by a ...
0
votes
0answers
24 views

Notation in H-spaces from Homotopy point of view

This book doesn't have a notation index. What do m|e, and rel(e,e) likely mean? Many thanks!
0
votes
0answers
35 views

How many topologically inequivalent loops are there on genus g surface?

Suppose we had a 2D surface with g holes in it, and suppose a child drew closed loops on that surface. How many topologically distinct loops can be drawn on the surface? Two loops are equivalent if ...
4
votes
3answers
175 views

uses of Riemannian geometry for questions not related to Riemannian geometry

Poincaré conjecture (whose formulation does not make use of notions from Riemannian geometry: "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere") was eventually proved using ...
2
votes
0answers
146 views

Reference request: Tubular neighborhood theorem for non-closed manifolds via the exponential map.

Let $M\subset N$ be a submanifold. (Both $M$ and $N$ have no boundary, but $M\subset N$ need not to be closed as a subspace and none of them need to be compact) Choose a Riemannian metric on $N$ and ...