# homotopy

#### Stiger

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.

#### Drexel28

MHF Hall of Honor
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.
Is this topological dimension or dimensionality as a manifold? Is $$\displaystyle S^p=\mathbb{S}^p=\left\{\bold{x}\in\mathbb{R}^{p-1}:\|\bold{x}\|=1\right\}$$?

#### xxp9

sorry I was wrong

#### Stiger

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!!