Let $X$ be a path-connected topological space. And there is a continuous map $F: X\times X \to X$ such that: $$F(x,x)=x \ \text{ and }F(x,y)=F(y,x).$$ Prove: The fundamental group of $X$ is Abelian.

This problem is in my topology class homework. We can show that if f,g are loops in X, then F(f,g) is also a loop in X. Let [f] be the element in the fundamental group represented by f, and e be the constant map which represents the unit element in the fundamental group. Then for any [f] and [g], we have [F(f,e)][F(g,e)]=[F(g,e)][F(f,e)]. It turns out that [f]->[F(e,f)] is a group homomorphism. And the image of this homomorphism is an Abelian group. However, I cannot show that this homomorphism is surjective, which is needed to solve this problem.

Guys I've come up with a solution. Use [F(f,e)][F(g,e)]=[F(g,e)][F(f,e)] and x=F(x,x)=F(e,x)F(e,x).

  • $\begingroup$ I have shown that if for any loop f and g, F(e,f)*F(e,g)=F(e,g)*F(e,f). But I can not prove the map f->F(e,f) is surjective. $\endgroup$ – Li Xinghe Nov 21 '13 at 6:07
  • $\begingroup$ In your notation, can you explain what is e? $\endgroup$ – user99914 Nov 21 '13 at 6:14
  • $\begingroup$ e is the unit element in the fundamental group. $\endgroup$ – Li Xinghe Nov 21 '13 at 6:16

To see that $F_*$ is surjective, observe that since any curve $f$ in $X$ lifts to the curve $(f,f)$ in $X\times X$, any loop $[f]\in \pi_1(X)$ lifts to $([f],[f])\in \pi_1(X)\times\pi_1(X)\cong\pi_1(X\times X)$. Since $F(x,x) = x$, you know that $F_*([f],[f]) = [f]$.

  • $\begingroup$ Sorry Neal, could you give more detail? I'm afraid I can not follow you! How can you prove that [f][g]=F∗([f],[g])? $\endgroup$ – Li Xinghe Nov 21 '13 at 12:36
  • $\begingroup$ @LiXinghe I will add more when I'm at work. $\endgroup$ – Neal Nov 21 '13 at 12:59
  • $\begingroup$ Thanks, Neal. Meanwhile I'll keep on trying on it myself. $\endgroup$ – Li Xinghe Nov 21 '13 at 13:09
  • 2
    $\begingroup$ @LiXinghe You should post your solution as an answer to your own problem! $\endgroup$ – Neal Nov 22 '13 at 4:20

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