Hi,
I'm taking first steps in algebraic topology and I became stuck at the problem:
"Prove that any map of the real projective space RP^n for n>=2 to S^1 is null homotopic" ("Algebraic topology" by E. Spanier. Ex.4 chapter2)
I've already proved that any two maps from a simply connected locally path-connected space to S^1 are homotopic (Ex.1,2,3 from same chapter).
Below I sketch what I have done. (However, I'm afraid my path isn't homotopic with a solution ;| ).
S^n is (simply connected) double cover of RP^n . From the above we have that any two maps from S^n to S^1 are homotopic. In particular, for an arbitrary function g:S^n -> S^1 there exists a homotopy
H: S^nxI -> S^1 such that:
H(x,0)=g(x) for all x in S^n
H(x,1)=p for all x in S^n (p is some fixed point from S^1)
Now, for an arbitrary function f: RP^n -> S^1 I define a homotopy
F: RP^nxI -> S^1 by:
F(x,t)=H(y,t) where:
H is a homotopy given above where g is a composition of a covering projection c: S^n -> RP^n with function f. As c is a double cover I need to specify which point y choose to compute H(y,t). (for RP^2 it is quite easy, but what for n greater than 2?)
Looking forward for any clues and comments
magda