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What is a loop in $\mathbb RP^n$ ?

I have to show that:

Given a loop $\alpha:[0,1]\rightarrow\mathbb RP^n$ starting and ending in $x_0=[N]=[S]$ and its lift $\tilde{\alpha}:[0,1]\rightarrow S^n$ starting from $N$

Prove that;

$\chi:\pi_1(\mathbb RP^n, x_0)\rightarrow\{-1,1\}$ with;

$\chi([\alpha])=1,if\ $$\tilde{\alpha}[1]=N$ or

$\qquad$$\quad$$-1,if\ $$\tilde{\alpha}[1]=S$

is a group homomorphism

where N=(1,0,0...), S=(-1,0,0...)

EDIT:

first i have to show that the map is well-defined, i.e. $\alpha\simeq\alpha'\Rightarrow\tilde{\alpha}(1)=\tilde{\alpha'}(1)$

but, when are $\alpha$ and $\alpha'$ homotopic in $RP^n$ ?

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derivative
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  • A loop is a closed path. A continuous $\alpha \colon [0,1]\to X$ with $\alpha(0) = \alpha(1)$. Or, equivalently, a continuous $\alpha \colon S^1 \to X$. – Daniel Fischer Feb 12 '14 at 13:11
  • @Daniel Fischer for example is the half circle closed in $RP^2$ ? – derivative Feb 12 '14 at 13:13
  • Which half-circle? A half-circle in $S^n$ is projected (under the canonical map) to a loop in $\mathbb{R}P^n$. The exercise speaks of loops in $\mathbb{R}P^n$ and their lifts in $S^n$. The lift of a loop is not necessarily a loop. – Daniel Fischer Feb 12 '14 at 13:15
  • no, i made just an example in 2-dim case. – derivative Feb 12 '14 at 13:17
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    If two loops are homotopic, their lifts with the same initial point have the same end point. And conversely. – Daniel Fischer Feb 12 '14 at 13:29

1 Answers1

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A loop in $\mathbb{R}P^n$ is a closed path $\alpha\colon[0,1]\to \mathbb{R}P^n$. That is, $\alpha(0)=\alpha(1)$. By the path lifting theorem, we can equivalently say that if $f\colon S^n\to\mathbb{R}P^n$ is a two-fold covering map with $f(N)=f(S)$ then a path $\alpha \colon [0,1] \to \mathbb{R}P^n$ with $\alpha(0)=[N]$ is a loop if and only if the induced lift (based at $N$) $\tilde{\alpha}\colon[0,1]\to S^n$ along $f$ is a path with either $\tilde{\alpha}(1)=N$ or $\tilde{\alpha}(1)=S$.

Note that if $\tilde{\alpha}(1)=N$ then $\tilde{\alpha}$ is a loop in $S^n$ and so it is true that the image of any loop in the total space of a covering map is also a loop in the base space. It is not true that the preimage of a loop under a covering map is also a loop however. The most obvious example would be a loop with non-zero winding number in the circle and the corresponding lift under the exponential map $\exp\colon\mathbb{R}\to S^1$.

Dan Rust
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