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I'm trying to gain intuition on how the projective space fundamental group and other algebraic topology concepts work, as in this question.

I know that the fundamental group of $\mathbb{RP}_2$ is $\mathbb{Z}_2$. This means that one can glue homeotopic loops as one sums elements in $\mathbb{Z}_2$. Now let me ask you how do this loops look in different models of the projective space:

Antipode model

In this model one identifies opposite elements of $\mathbb{S}^2$ and obtains a quotient space (which honestly I cannot imagine very well) which is homeomorphic to $\mathbb{RP}^2$. What are the two classes of loops that one can identify in $\mathbb{S}^2$ that give the two distinct homeotopically equivalent classes of loops that one finds in the fundamental group? How does the operation of this loops relate to the operation in $\mathbb{Z}_2$?

Point of infinity model

This is the model that I studied in projective geometry. One sets a normal plane and adds a parallel external plane which is supposed to represent a "line of infinity" the rest of lines are intersections of external planes with the normal plane.

How do loops of the fundamental group look like in this model? How do you operate with them.

My thoughts.

Now that I review it, clearly the zero element must be a constant loop. It remains to interpret the non-constant loop.

Seeing the answer I think I understand how the thing works in the identified sphere. There one can deform the whole loop into the trivial one easily. But for the loop that goes from antipode-antipode one cannot deform it into anything else that is not another loop antipode-antipode since are antipodes are identified as the same point. However I wonder what would happen if they were deformed into the very same point (that is not continuous?)

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    $\begingroup$ You have a loop in $\mathbb{RP}_2$, it lifts to either a loop in $\mathbb{S}^2$ or a path from the starting point to its antipode. The first represents the trivial element in $\pi_1(\mathbb{RP}_2)$, the second is the nontrivial element. $\endgroup$ – Bob Jones Dec 3 '17 at 0:04
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In the antipode model, the non-contractible classes of loops are the ones which connect a point on the sphere to its antipode.

In the plane-union-line-at-infinity model, a standard line, e.g., $y=mx$, together with its single limit point at infinity, forms a non-contractible loop.

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  • $\begingroup$ how do you see the fact that for instance in the antipode model i cannot deform the non-trivial loop into the trivial one? $\endgroup$ – Rodrigo Dec 3 '17 at 12:14
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    $\begingroup$ @Rodrigo it's precisely because the nontrivial loop doesn't lift to a loop on $S^2$ at all; it lifts to a path connecting a point to its antipode. Deforming such a path in a way that preserves the endpoints won't change the fact that the endpoints don't meet. $\endgroup$ – Dustan Levenstein Dec 3 '17 at 15:44

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