What Determines the Degree of a Continuous Map Between Spheres?

[quasar987]

(1) For a continuous map f:S^n-->S^n, the induced map in top homology is, up to identification of H_n(S^n) with Z, just multiplication by an integer, and this integer is defined as the degree of f. (in Hatcher: http://www.math.cornell.edu/~hatcher/AT/ATpage.html)

Now, in the event that f is a homeomorphism, it follows from the elementary properties of the degree that deg(f)=±1. After this remark, Hatcher adds that in applications, it is usually not hard to determine which it is between +1 and -1.

Can someone give an example illustrating how one decides between +1 and -1?


(2) How to see why the above definition of degree coincide with the one in terms of preimage of regular value in the case of a differentiable f?

Thanks.

 

[wofsy]

(1) For a continuous map f:S^n-->S^n, the induced map in top homology is, up to identification of H_n(S^n) with Z, just multiplication by an integer, and this integer is defined as the degree of f. (in Hatcher: http://www.math.cornell.edu/~hatcher/AT/ATpage.html)

Now, in the event that f is a homeomorphism, it follows from the elementary properties of the degree that deg(f)=±1. After this remark, Hatcher adds that in applications, it is usually not hard to determine which it is between +1 and -1.

Can someone give an example illustrating how one decides between +1 and -1?(2) How to see why the above definition of degree coincide with the one in terms of preimage of regular value in the case of a differentiable f?

Thanks.

Give the sphere an orientation. A smooth homeomorphism that preserves orientation will have degree one. If it reverses orientation it will have degree minus one. This you can tell from the determinant of the Jacobian at any point where the Jacobian has maximal rank.

If the map is not differentiable but only continuous it may be difficult to tell.
In general you need to follow the fundamental cycle,C, as it is mapped into the sphere
to the cycle, f(C), and decide whether C-f(C) is a boundary.

If two differentiable maps are homotopic then they have the same degree. This is always true.

But for spheres the converse is also true. It two maps from a manifold into a sphere have the same degree then they are homotopic.

Good examples of maps of arbitrary positive degree on S^2 are complex polynomials on the Riemann sphere.