degree of a map between Manifold and Hopf theorem
Let $M$, and $N$ be closed orientable manifolds with the same dimensions. Hopf Proved that if $N=mathbb{S}^n$, then every two continuous maps from $M$ to $N$ with the same degree are homotopy equivalence. In particular, each degree zero map from $M$ to $mathbb{S}^n$ is null-homotopic.
$textbf{Q})$ When a degree zero map between closed orientable manifolds with the same dimensions is a null-homotopy? Or, is there any generalization of Hopf theorem for the case that the target space is not a sphere?
algebraic-topology
asked Dec 12 '18 at 7:03
123...123...
416213
add a comment |
Let $M$, and $N$ be closed orientable manifolds with the same dimensions. Hopf Proved that if $N=mathbb{S}^n$, then every two continuous maps from $M$ to $N$ with the same degree are homotopy equivalence. In particular, each degree zero map from $M$ to $mathbb{S}^n$ is null-homotopic.
$textbf{Q})$ When a degree zero map between closed orientable manifolds with the same dimensions is a null-homotopy? Or, is there any generalization of Hopf theorem for the case that the target space is not a sphere?
algebraic-topology
asked Dec 12 '18 at 7:03
123...123...
416213
Unfortunately there is no such generalization.
– Mike Miller
Dec 14 '18 at 15:31
add a comment |
Let $M$, and $N$ be closed orientable manifolds with the same dimensions. Hopf Proved that if $N=mathbb{S}^n$, then every two continuous maps from $M$ to $N$ with the same degree are homotopy equivalence. In particular, each degree zero map from $M$ to $mathbb{S}^n$ is null-homotopic.
$textbf{Q})$ When a degree zero map between closed orientable manifolds with the same dimensions is a null-homotopy? Or, is there any generalization of Hopf theorem for the case that the target space is not a sphere?
algebraic-topology
asked Dec 12 '18 at 7:03
123...123...
416213
Let $M$, and $N$ be closed orientable manifolds with the same dimensions. Hopf Proved that if $N=mathbb{S}^n$, then every two continuous maps from $M$ to $N$ with the same degree are homotopy equivalence. In particular, each degree zero map from $M$ to $mathbb{S}^n$ is null-homotopic.
$textbf{Q})$ When a degree zero map between closed orientable manifolds with the same dimensions is a null-homotopy? Or, is there any generalization of Hopf theorem for the case that the target space is not a sphere?
algebraic-topology
algebraic-topology
asked Dec 12 '18 at 7:03
123...123...
416213
asked Dec 12 '18 at 7:03
123...123...
416213
asked Dec 12 '18 at 7:03
123...123...
416213
asked Dec 12 '18 at 7:03
123...123...
416213
asked Dec 12 '18 at 7:03
123...123...
416213
416213
Unfortunately there is no such generalization.
– Mike Miller
Dec 14 '18 at 15:31
add a comment |
Unfortunately there is no such generalization.
– Mike Miller
Dec 14 '18 at 15:31
Unfortunately there is no such generalization.
– Mike Miller
Dec 14 '18 at 15:31
Unfortunately there is no such generalization.
– Mike Miller
Dec 14 '18 at 15:31
add a comment |
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Unfortunately there is no such generalization.
– Mike Miller
Dec 14 '18 at 15:31