7
votes
0answers
128 views

If a thread is pulled out of a floating blob of water, must the thread be tangent to the surface of the blob at some point?

My motivation is the recent question I just answered, and my answer use too many hypothesis that I considered superfluous: Always "one double root" between "no root" and "at ...
2
votes
0answers
61 views

Role of the Thom space in the Pontryagin-Thom construction

I am trying to understand the Pontryagin-Thom theorem; especially how the Thom space comes into play. Just to bring everyone on the same page: I am specifically talking about the construction of an ...
2
votes
1answer
52 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 ...
1
vote
0answers
68 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
39 views

Book of Pullbacks and Pushouts

what books can I consult for properties of pullback and pushouts in algebraic topology? I need to understand the theory of homotopy in algebraic topology and I started to study pullbacks and push ...
0
votes
0answers
62 views

Will the pullback of homotopic maps give rise to isomorphic fibre bundles?

I know it's certainly right for the case of vector bundles, but what about fibre bundles?
0
votes
1answer
44 views

Does the structure group of S^n homotopic to O(n+1)?

It is easy to show that Diff(S^1) is homotopic to O(2), but in the case of n bigger than 1 things become really complicated, I cannot see the conclusion directly.
1
vote
0answers
41 views

Computing group of exotic spheres

In Levine on page 90 it is stated that the following sequence is exact $$ 0 \to bP^{n+1} \to \Theta^n \to Coker(J_n) $$ where $\Theta^n$ is the group of exotic spheres, $bP^{n+1}$ is the subgroup of ...
2
votes
0answers
61 views

What happens in dimension 125?

In Differential topology 46 years later (page 807, bottom of left column) Milnor states that for $n \neq 4, 125, 126$ if the order of the stable homotopy groups $|\Pi_n|$ is known then we can compute ...
2
votes
1answer
103 views

Exotic spheres and homotopy groups: sanity check

There is a homomorphism $\Theta_n \to \Pi_n/J_n$ where $\Theta_n$ is used to denote the group of diffeomorphism classes of $n$-spheres (with connected sum), $\Pi_n$ denote the $n$-th stable homotopy ...
3
votes
1answer
54 views

$J(X)$ and exotic spheres.

I read that we can realize exotic sphere as the coker of $J$-homomorphism. SO we can consider the exotic sphere $S^7$ realized using an identificantion of $B^4 \times S^3$ (where I donote the four ...
3
votes
1answer
63 views

How to find stable homotopy group given the quotient group?

If $\Theta_n$ is the group of exotic spheres in dimension $n$ and $\mathrm{bP}_{n+1}$ is the group of spheres that bounds parallelizable $(n+1)$-manifolds, $\pi_n^S$ is the $n$th stable homotopy group ...
3
votes
0answers
102 views

Are strongly close maps homotopic?

While reading about various results related to density of smooth functions in the space of continuous functions with strong topology, I've got the impression that it is a general fact that for any ...
14
votes
1answer
188 views

Are locally homotopic functions homotopic?

Suppose we have two (smooth) functions $f,g:X\to Y$, where $X,Y$ are smooth (second-countable, Hausdorff) manifolds which are locally homotopic (that is, any point in $X$ has a neighbourhood $U$ such ...
1
vote
0answers
64 views

How to prove this isotopy exist?

Let $M$ be a topological manifold and $N$ is a subset of $M$. Let $f_t$ be an isotopy from $id $ to $f$ which is a homeomorphism on $M$. Suppose $f(N)=N$ and there is an isotopy $g_t$ on $N$ such that ...
3
votes
1answer
51 views

Are these functions homotopic?

Let $\gamma$ be a smooth, simple, closed curve and let $f : \gamma \to S^1$ assign to each $x \in \gamma$ the unit normal vector there. We can find a diffeomorphism $g: \gamma \to S^1$ and define the ...
3
votes
1answer
95 views

Suspension Operation on the Pontryagin-Thom Construction

I have a feeling that this is well-known: View the Pontryagin-Thom construction as the bijective correspondence between $[M,S^r]$ and the set of (appropriate equivalence classes of) framed ...
9
votes
1answer
124 views

Are close maps homotopic?

Consider a smooth manifold $M = M^m$ and a smooth submanifold $N = N^n \subset M$. Suppose that two maps $f, g: M \to N$ are close to each other, in the sense that there exists $\epsilon > 0$ such ...
10
votes
1answer
495 views

If $f\!: X\simeq Y$, then $X\!\cup_\varphi\!\mathbb{B}^k \simeq Y\!\cup_{f\circ\varphi}\!\mathbb{B}^k$.

How can I prove that if two spaces $X$ and $Y$ are homotopy equivalent, then the corresponding spaces obtained by gluing a $k$-cell are also equivalent? In detail, if ...
6
votes
1answer
414 views

Continuous maps between compact manifolds are homotopic to smooth ones

If $M_1$ and $M_2$ are compact connected manifolds of dimension $n$, and $f$ is a continuous map from $M_1$ to $M_2$, f is homotopic to a smooth map from $M_1$ to $M_2$. Seems to be fairly basic, ...
8
votes
1answer
801 views

Diffeomorphism group of the unit circle

I am given to understand that the group of diffeomorphisms of the unit circle, $\operatorname{Diff}(\mathbb{S}^1)$, has two connected components, $\operatorname{Diff}^+(\mathbb{S}^1)$ and ...
2
votes
2answers
215 views

Degree of a map between product of manifolds

Let $M^m$ and $N^n$ be compact, oriented smooth manifolds without boundary. Then what is the degree of the map $$ f: M\times N \to N \times M$$ given by $f(x,y) = (y,x)$? I have the feeling it ...