homotopy

Dec 2008
16
0
If \(\displaystyle dimM=m<p\), show that every map(maybe continuous) \(\displaystyle M^m \to S^p\) is homotopic to a constant.


This is the problem 5 in chap8. of 'topology from the differentiable viewpoint(Milnor)'.


I proved it when the map is not onto. But I think it can be onto.
Please help me.
 

Drexel28

MHF Hall of Honor
Nov 2009
4,563
1,566
Berkeley, California
If \(\displaystyle dimM=m<p\), show that every map(maybe continuous) \(\displaystyle M^m \to S^p\) is homotopic to a constant.


This is the problem 5 in chap8. of 'topology from the differentiable viewpoint(Milnor)'.


I proved it when the map is not onto. But I think it can be onto.
Please help me.
Is this topological dimension or dimensionality as a manifold? Is \(\displaystyle S^p=\mathbb{S}^p=\left\{\bold{x}\in\mathbb{R}^{p-1}:\|\bold{x}\|=1\right\}\)?
 
Mar 2010
293
91
Beijing, China
sorry I was wrong
 
Dec 2008
16
0
Is this topological dimension or dimensionality as a manifold? Is \(\displaystyle S^p=\mathbb{S}^p=\left\{\bold{x}\in\mathbb{R}^{p-1}:\|\bold{x}\|=1\right\}\)?
Yes! You're right!!