#projectivespace

Smoke grey vibes w @Kirstenchilstrom #Fashion #Work #Modelsdot #ChristopherMcCartha #RedModels #Projectivespace (at Projective Space)

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Come and join amazing people, sharing their awesome projects, this Tuesday at Projective. http://jam.sh/23vbOYW

http://jam.sh/1YAZ5lU

We turn to \(\mathbb{CP}^1\). Things are less visually clear here, since we are looking at lines in \(\mathbb{C}^2\), which has real dimension 4 and is, therefore, much harder to directly visualise. We will proceed by constructing a map from the set of lines to points of the form \( (z,1) \), with \( z \in \mathbb{C} \). Let \([z_0,z_1]= \{ (tz_0, tz_1): t \in \mathbb{C} \} \) be such a line. Then map the line to the point \( (z_0/z_1, 1) \), where \(t= 1/z_1\). This is injective since \( z_1/z_2=z_0/z_1 \) implies \( (z_1, z_2) \in [z_0,z_1] \). Further, let \((z,1)\) be arbitrary, then we can construct the appropriate line \([z,1]= \{ (tz, t): t \in \mathbb{C} \}\) so our map is surjective. We can also check that it is a homeomorphism using the fact that \(\mathbb{CP}^1\) is a quotient space. Note that our map doesn’t work for the line \([w,0]\) because we can’t divide by zero. This line is dealt with by mapping it to the point at infinity. Filling in some gaps, this shows us that \(\mathbb{CP}^1\) is diffeomorphic \(S^2 \cong \mathbb{C}_\infty \). What we have done is something like parametrising the set of (real) lines in \( \mathbb{R}^3 \) by their point of intersection with the plane \(z=1\).

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