# Show that every map(maybe continuous)

If dimM=m<p, show that every map(maybe continuous) Mm -> Sp 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.

quasar987
Homework Helper
Gold Member

It can indeed be onto. For instance, for M=S^1 and S^p=S², take f:S^1-->[0,1]² one of the infamous space-filling curve (loop) (http://en.wikipedia.org/wiki/Space-filling_curve). Then make [0,1]² into S² by identifying all the edges together. Then p o f:S^1-->S² is a continuous surjection, where p:[0,1]²-->[0,1]²/~=S² is the quotient map.

Office_Shredder
Staff Emeritus
Gold Member

If dimM=m<p, show that every map(maybe continuous) Mm -> Sp 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.