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
votes
1answer
75 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
41 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
262 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
294 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
179 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
55 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
62 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 $\mathbb{R}^3$?
0
votes
1answer
12 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
65 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
187 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 ...
8
votes
1answer
125 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
122 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
152 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
53 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 $\mathbb{R}^{n+1}...
7
votes
1answer
51 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
77 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 component....
0
votes
1answer
62 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
90 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 $M_\...
1
vote
1answer
76 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
166 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
53 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
69 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
56 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
65 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
101 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
66 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
226 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
134 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
259 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
18 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
63 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 $B^n$,...