0
votes
0answers
19 views

Link complement simply-connected if codimension $\geq 3$

In Rolfsen, page 50 says that "an easy general position argument shows that a PL link $L^k$ in $S^n$ has simply-connected complement if $n - k > 3$," where $L^k$ is a $k$-dimensional link in $S^n$. ...
-2
votes
0answers
50 views

Why Must the Degree of this Map be 0? [on hold]

Let $f:S^3 \rightarrow S^1\times S^1\times S^1$ be a continuous map. Show that it's degree must be $0$. (Just a hint would be good)
2
votes
2answers
55 views

Embedding of $\mathbb{R}^2 \to \mathbb{R}^3$ with non-parallel tangent planes

I have a qual question here and I'm struggling to get a good starting point. The question asks to construct a smooth proper embedding $f\colon \mathbb{R}^2 \to \mathbb{R}^3$ such that for any distinct ...
1
vote
2answers
84 views

Why are open sets used in definitions in differential geometry?

I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are ...
2
votes
1answer
85 views

$\mathbb{R}^2$ is a Retract of $\mathbb{R}^5$

Looking through some old qualifying exams, I noticed the question: Let $X\subseteq \mathbb{R}^5$ be homeomorphic to $\mathbb{R}^2$. Prove that $X$ is a retract of $\mathbb{R}^5$. I'm not even sure ...
2
votes
1answer
50 views

Stability of the nonempty intersection of an open set $A$ with a set $S$ under homotopy?

To be more precise: let $F(x,t) : R^2 \times I \to R^2$ be a homotopy of open maps $F(_,t)(x)$ (the restriction of $F$ to some fixed $t$) (the homotopy is continuous in both variables). Suppose that ...
3
votes
1answer
80 views

Real analysis with a non-standard topology

I have recently undertaken a self study of topology and am using Munkres Topology 2nd edition as the primary text. My background(theoretical chemistry & physics) is almost entirely void of any ...
1
vote
0answers
33 views

Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
8
votes
1answer
208 views

Surgery results in a cylinder

While reading a proof of a theorem about Reshetikhin Turaev topological quantum field theory, I encountered the following problem. Suppose we have several unlinked unknots $K_i$, $i=1, \dots, g$ in ...
1
vote
1answer
30 views

Closed regular neighborhood

I would like to understand the following sentences. Let $L$ be a framed link in the three dimensional sphere $S^3$. Suppose $L$ has $m$ components $L_1, \cdots, L_m$. Let $U$ be a closed regular ...
0
votes
1answer
25 views

Euler class is odd under orientation, thus its integral over a manifold will be even.

I learned a statement from others: "Euler class is odd under orientation, thus its integral over a manifold $M$ is even." I cannot fully appreciate it, can someone show this explicitly? The ...
1
vote
1answer
136 views

two interlocked circles are homeomorphic to two noninterlocked circles

This is what I learned from here the post: two interlocked circles are homeomorphic to two noninterlocked circles, thus they (two interlocked circles and two noninterlocked circles) are homotopic ...
1
vote
2answers
150 views

Definition of “a topological manifold with corners”.

How can we define a topological manifold with corners and its corners? Then, do we use "invariance of domain" to define corners, as we really need this theorem in order to define "boundaries of a ...
2
votes
1answer
65 views

Why surgery produce a new 3-manifold?

I was studying a proof of the fact that any closed orientable 3-manifold is obtained by integer surgery along a link. I read the several proofs but I don't understand well. A proof is as follows. ...
1
vote
2answers
84 views

Homology of manifolds with boundary

If $M$ is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is ...
1
vote
0answers
63 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 ...
0
votes
0answers
32 views

Definition of Jets

Can someone help me with a definition of jets between Cr manifolds. I want to avoid using inverse limit at infinity, but how do we define jets then ?
1
vote
0answers
27 views

Is the $C^r(M, N)$ space, with the strong (Whitney) topology, a Fréchet-Urysohn space?

Given smooth, non-compact manifolds $M$ and $N$, consider the function space $C^r(M, N)$. Equipped with the strong (Whitney) topology, this space is Hausdorff and Baire. It is, however, not first ...
0
votes
0answers
14 views

Calabi homomorphism of the disk

There is a fact that the homomorphism $Diff_0^{\infty}(\mathbb{D},\partial\mathbb{D},area)\to \mathbb{R}$ is surjective, we can use Calabi homomorphism to prove it, where ...
0
votes
0answers
24 views

History of vectorial bundles in articles or papers?

I'm looking for an article or book that gives a thorough and interesting history of bundles and vectorial bundles in algebraic topology. I'm looking for it for my own learning, please help its ...
2
votes
1answer
53 views

Definition for Euler characteristic without CW-complexes

It is possible to have a definition of the Euler characteristic without using CW-complexes? (I'm referring to the definition given by Wikipedia : ...
2
votes
1answer
68 views

Contractibility of the space of sections of a fiber bundle

Let $\pi: E \to M$ a fiber bundle and $\Gamma(M,E)$ the space of smooth sections of the bundle with topology induced by the Whitney topology on $C^{\infty}(M,E)$. Assume that each fiber is ...
0
votes
1answer
81 views

compact open topology on M(X,Y) continuous function space

I'm really confused with the statement. let $X$ be a set endowed with the discrete topology and let $Y$ be any topological space. Show that $M(X,Y)$ with the compact open topology is homeomorphic to ...
6
votes
2answers
144 views

Prerequisites for Freedman's proof of the 4-dimensional Poincaré conjecture

I have a good understanding of differential geometry, enough at least to understand many details of Hamilton & Perelman's approach to the 3-dimensional Poincaré conjecture. I have no such ...
0
votes
1answer
16 views

If two maps' derivatives have unit length, then the derivative of the product is $\pm 1$

Let $M$ be a space and $I$ the unit interval. Definition A map $f : I \to M$ is a parametrization by arc-length if $f$ maps $I$ diffeomorphically onto an open subset of $M$, and if the "velocity ...
10
votes
3answers
407 views

Why care about group actions?

Let X be a space (topological space, manifold, etc) and let the group G act continuously on X. What extra (homotopical, homological, cohomological, diffeomorphical etc) data can extracted from X when ...
4
votes
1answer
115 views

Turn a torus inside out

Let $T=D^2 \times S^1$ be a solid torus, where $D^2$ is a 2-dimensional disk and $S^1$ is a circle. Suppose we have another solid torus $T'$ and we have a homeomorphism $f$ sending a meridian ...
4
votes
2answers
123 views

Any N dimensional manifold as a boundary of some N+1 dimensional manifold?

Is this statement true: Question: Can any N dimensional manifold be realized as a boundary of some N+1 dimensional manifold? If so/not, how to prove/disprove it? I read a TQFT paper from Edward ...
1
vote
1answer
54 views

The boundary of this set is smooth?

Let $\Omega_1 \supset \Omega_2 \supset....$ a decreasing sequence of bounded, convex and smooth sets. My intuition says that the set $int(\overline{\bigcap_i \Omega_i})$ (where int denotes the ...
1
vote
1answer
36 views

Extension of funcion

I think its right but Im not sure. I have topological space (exactly manifold - second countable, Hausdorff, local Euclidean topological space) M, dim M=m. Let $A \subset M$ is closed set, dim A=n, ...
5
votes
0answers
93 views

Cohomology of covering space

Let $B$ be a base space and $E$ be a covering space of $B$ what is the relation between $H^2(B,\mathbb{Z})$ and $H^2(E,\mathbb{Z})$.?
0
votes
0answers
54 views

Chart on a manifold

I have the following question. If I consider a manifold, for example a torus T see as space of identification $[0,1]\times [0,1]$ why I can't cover it with only one chart? what fails if a chart ...
1
vote
1answer
27 views

Involutive Properties of Space-structures on Smooth Manifolds

I am currently reading Quantum Invariants of Knots and 3-Manifolds by Turaev, and I am having a hard time understanding a statement made on page 120. He is explaining the property of space-structures, ...
0
votes
1answer
58 views

On the “regularity” of the boundary of an open set

Let $M = \mathbb{R}^2$ (or more generally, let $M$ be a topological manifold) and let $\Omega$ be an open set in $M$. I'm considering the following regularity conditions for the boundary of $\Omega$: ...
1
vote
1answer
66 views

Product manifolds

I have a question on the product of two manifolds. I have $M, N$ two real manifolds (with a smooth differentiable structure), with $\partial M=0$. I have showed that $M\times N$ has a natural induced ...
2
votes
2answers
57 views

homeomorphism between maninifolds

Exist a local homeomorphism between the manifolds with boundary $[0,1) \times [0,1) $ and $\mathbb{R}^{2}_{+}$? I don't think that a local homeomorphism like this can exist..
3
votes
0answers
53 views

Crosscap function in $\mathbb{R}^4$ - and how to show it is proper?

I found the Cross-cap function in $\mathbb{R}^3$ as follows: $$f(x,y,z)=(yz,2xy,x^2-y^2),$$ My questions are (I couldn't show any progress for Q1,2.I have thought hard but had no clue): Q1: Is ...
0
votes
1answer
57 views

$GL(n,\mathbb{C})$ is semi-locally simply connected.

How can we show that $GL(n,\mathbb{C})$ is semi-locally simply connected.
1
vote
1answer
29 views

$dg_y$ carries a subspace of $T_y(Y)$ onto $\mathbb{R}^l$

The question arises from Guillemin and Pallack Page 28 above the frame: $dg_y$ carries a subspace of $T_y(Y)$ onto $\mathbb{R}^l$ precisely if that subspace and $T_y(Z)$ span all of $T_y(Y)$. I ...
3
votes
1answer
55 views

Are space of paths between two different points and space of pointed loops only homotopy equivalent? What about smooth case?

Let $X$ be a path-connected CW-complex and $x$, $y$ points in $X$. Any choice of a path between $x$ and $y$ provides maps (in both directions) between the space $L(x, y)$ of paths from $x$ to $y$ and ...
2
votes
1answer
47 views

Topological property of some manifold

I am provided with a smooth map $g : N \rightarrow N^\prime $ between differentiable manifolds. $N$ is assumed to be compact and connected. Moreover, the differential $dg_x: T_x N \rightarrow ...
0
votes
1answer
39 views

Assumptions required for an implicitely defined surface/manifold to have a specified dimension

What are some normal assumptions made on implicitly defined manifolds? More specifically, by implicitly defined manifold, I mean the definition of a surface such as $g(x,y)=x^2+y^2-1=0$ for the ...
1
vote
0answers
32 views

Cylinder as Fibre bundles

I have to show that the cylinder C is a fibre bundle over $S^1$ with fibre an open interval and I have to write a trivialization and the cocycles. I think that this is a trivial bundle, because I can ...
3
votes
2answers
104 views

How does one prove that the Klein bottle cannot be embedded in $R^3$?

How does one prove that the Klein bottle cannot be embedded in $R^3$? I'm talking about smooth embeddings.
2
votes
3answers
75 views

Topological spaces with prescribed fundamental groups

The question I am about to ask could have gone to the chat section but I want to have the answers/comments in an easy-to-refer-back-to style. For (connected, pointed) topological spaces with trivial ...
2
votes
1answer
69 views

Is the map from the circular half cone to the $xy$ plane a local isometry?

This is a text book exercise. And I think that this map is not a local isometry. But, I don't know how to show this question. Please help me explaining this question. Thanks a lot. I posted its ...
1
vote
1answer
58 views

The question related to a regular surface.

Prove that an equation of the form $f(x,y,z)=c$ determines a regular surface if $f$, defined on some open subset $S$ of $\mathbb{R}^3$, is smooth and $\nabla f\neq 0$ everywhere in $S$. I know ...
1
vote
0answers
90 views

prove that $f$ is a diffeomorphism and an isometry

Let $S_1 : [0, 2\pi r]\times [0, h]$ $S_2: x^2+y^2=r^2$ Let $f: S_1 \to S_2$ $(u,v)=(r\cos (\frac{u}{r}), r\sin (\frac{u}{r}), v)$ for $v\in [0,h]$ and $u\in [0, 2\pi r)$ How do I prove that ...
0
votes
1answer
61 views

Verify this is not orientable.

Verify this is not orientable. Möbius transformation: $$U=\{(t,\theta) \mid \frac{-1}{2}\lt t\lt \frac{1}{2}, 0\lt \theta \lt 2\pi \}$$ $\sigma (t, \theta)=<((1-t\sin (\theta/2))\cos (\theta), ...
0
votes
0answers
55 views

What is the type of $u$ in this definition of knot?

I am going to quote from the second paragraph of the Introduction of Möbius Energy of Knots and Unknots by Michael H. Freedman, Zheng-Xu He and Zhenghan Wang. Let $\gamma = \gamma (u)$ be a ...