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user 1465
5
votes
Mathematical value of constructing sphere eversions
There a variety of answers to your question. I have a particular bias. Let me transpose your question to another context.
Say someone comes up with a "computation" of the unstable homotopy groups …
- 44.3k
7
votes
Accepted
About the commutativity of the $1^\text{st}$ homotopy group of the space of knots
There is a fairly clean statement about which components of the "long knot space" have abelian fundamental group.
$$K_{3,1} = \{ f : \mathbb R \to \mathbb R^3 : f(t)=(t,0,0) \text{ for } |t|>1 \} \sub …
- 44.3k
4
votes
Is Cohen immersion conjecture (theorem) known for vector bundles?
The answer is no.
Perhaps the simplest counter-example is for vector bundles over $0$-manifolds. They all immerse in $\mathbb R^n$ where $n$ is the dimension of the bundle. This is a considerably b …
- 44.3k
7
votes
Accepted
Can a nontrivial $n$-sphere bundle over $M$ embed in $M\times \mathbb{R}^{n+1}$?
The idea is to use the straight-line homotopy to construct such an extension, and then perturb the homotopy to be a family of embeddings. …
- 44.3k