# algebraic topology – Show element of fundamental group is nontrivial

I’m learning cohomology, and I’d like to show the following:

Let $$i:mathbb RP^1tomathbb RP^n$$ be the usual embedding taking $$(x_0,x_1)mapsto(x_0,x_1,0,dots,0)$$, where $$nge2$$. Further, let $$v:S^1tomathbb RP^1$$ be the fibration $$(x_0,x_1)mapsto(x_0,x_1)$$. Show that $$(icirc v)$$ is a nontrivial element of $$pi_1(mathbb RP^n,*)$$.

I tried proof by contradiction: Suppose $$icirc v$$ is homotopic to the constant map $$c$$ which sends everything to $$*$$. Then these induce the same maps in cohomology: $$v^*circ i^*=(icirc v)^*=c^*$$. Recall that $$H^q(mathbb RP^m;mathbb Z/2mathbb Z)=mathbb Z/2mathbb Z$$ for all $$0le qle m$$. Let $$Omega_nin H^1(mathbb RP^n;mathbb Z/2mathbb Z)$$ be the nonzero element, and similarly define $$Omega_1in H^1(mathbb RP^1;mathbb Z/2mathbb Z)$$. Then I already know that $$i^*(Omega_n)=Omega_1$$. The Gysin sequence shows that $$v^*(Omega_1)=0$$.

I wanted to show that $$c^*(Omega_1)$$ is nonzero. But I feel like I don’t quite understand cohomology groups/rings enough. If we write $$Omega_n=text{cls}~omega_n$$, where $$omega_nin Z^1(mathbb RP^n;2)$$ is a homomorphism $$C_1(mathbb RP^n)tomathbb Z/2mathbb Z$$, then the goal is to show that $$omega_nc_#:C_1(S^1)to C_1(mathbb RP^n)tomathbb Z/2mathbb Z$$ is not a coboundary, but I haven’t been able to do this. After all, isn’t $$omega_nc_#$$ just the zero map?

This is Exercise 12.24 in Rotman, and includes the hint that $$pi_1(mathbb RP^n,*)congmathbb Z/2mathbb Z$$ and $$mathbb RP^1approx S^1$$. But I didn’t use either one.

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