0

Prove that any continuous map $f\colon RP^3\to S^1$ is homotopic to a constant map?

How to prove it? Thanks in advance.

Lee Mosher
  • 120,280
Baraa Tito
  • 41
  • 5
  • Huh? Any constant map is clearly continuous and homotopic to a constant map. Maybe you mean not homotopic to a constant map? – user10354138 Feb 07 '21 at 16:40
  • 1
    I edited your question by a missing verb "is", in order to fix the grammar and to say what I think you intended to ask. If I'm wrong then feel free to rever the edit. – Lee Mosher Feb 07 '21 at 16:46
  • 3
    A hint: use arbitrary map lifting lemma w.r.t. the cover $\Bbb R\ni t\longmapsto \exp(2\pi it)\in \Bbb S^1$ with $\pi_1(\Bbb RP^3)=\Bbb Z_2$. Define a homotopy between the lift and constant map$(\Bbb R$ is contractible$)$, and then compose the above covering with this homotopy. – Someone Feb 07 '21 at 18:44
  • 4
    You might also know that the homotopy classes of maps from a CW-complex $X$ to $S^1$ are in bijection with $H^1(X;\mathbb Z)$ which is $0$ in your case. This is a theorem of Hopf, and can be seen as a particular case of $H^1$ being a functor representable by Eilenberg-MacLane spaces – Pierre Elis Feb 07 '21 at 18:49
  • 2
    @User873110 Why not an official answer? – Paul Frost Feb 07 '21 at 23:10
  • @PaulFrost: Because the question deserves to be closed. – Moishe Kohan Feb 12 '21 at 13:54

1 Answers1

2

Since $\pi_1(\Bbb RP^3)=\Bbb Z_2$ and $\pi_1(\Bbb S^1)=\Bbb Z$ the induced group homomorphism $f_*:\pi_1(\Bbb RP^3)\to \pi_1(\Bbb S^1)$ is the zero homomorphism.

Hence, by the arbitrary map lifting lemma we can lift the map $f$ w.r.t the covering $\Bbb R\ni t\longmapsto \exp(2\pi it)\in \Bbb S^1$. That is there is a map $\widetilde f:\Bbb RP^3\to \Bbb R$ with $\exp\big(2\pi i\widetilde f(x)\big)=f(x)$.

Note that to lift a map, we must ensure that the domain of the map is locally path-connected and connected. But here we are dealing with connected manifolds, so we do not need to worry.

Now, consider the homotopy $H:\Bbb RP^3\times [0,1]\ni (x,t)\longmapsto \exp\big(2\pi i (1-t)\widetilde f(x)\big)\in \Bbb S^1$.

Someone
  • 493