$Pic(X)$ vs. $CP^\infty$

Thank you to Edward Frenkel for kindly explaining the difference between $CP^\infty$ and $Pic(X)$ (both classifying spaces of line bundles), and to Qiaochu Yuan for explaining why on earth $CP^\infty$ is the moduli space of line bundles over a point. Any errors are mine, not theirs.

As we saw in a Precursor to Characteristic Classes, $CP^\infty$ is the classifying space of complex line bundles over $X$. $CP^\infty$ is, in some sense, the moduli space of line bundles over a point. There’s only one isomorphism class of line bundles over a point — but then this one line bundle has automorphism group $C^\times$ (which is homotopy equivalent to $U(1)$).

Allow me to introduce you to something that looks a LOT like $CP^\infty$. Continue reading $Pic(X)$ vs. $CP^\infty$

A Precursor to Characteristic Classes

I’ll assume that you know what a line bundle is, and are comfortable with the following equivalences; if you aren’t familiar with the notation in these equivalences, John Baez might help. Note that integral cohomology := cohomology with coefficients in $\mathbb{Z}$.

$U(1) \simeq S^1 \simeq K(\mathbb{Z}, 1)$

$BU(1) \simeq CP^\infty \simeq K(\mathbb{Z}, 2)$

The aim of this post is to give you a taste of the beautiful world of characteristic classes and their intimate relationship to line bundles via the concrete example of how the second integral cohomology group of a space is actually the isomorphism classes of line bundles over that space.

That’s right! $H^2(X; Z) \simeq$ the isomorphism classes of (complex) line bundles over X.  It is in fact, a group homomorphism — the group operations being tensor product of line bundles and the usual addition on cohomology. This isn’t something that I understood at first glance. I mean, hot damn, it’s unexpectedly rich. Continue reading A Precursor to Characteristic Classes

Maps of a Torus

In Segal’s Bourbaki talk on Elliptic cohomology, he mentions offhandedly that:

The set of pairs of commuting elements of a group $G$ are the set of homotopy classes of maps of a torus into $BG$.

Semon Rezchikov kindly explained this to me, and I found his explanation so simple and pleasing that I wish to share it.

Recall that $BG=K(G,1)$ and that $G = \pi_1(BG)$. A homotopy class of maps $S^1 \to BG$ is an element of $G$ (i.e. $BG$ is the delooping of $G$).

[If this statement confuses you, dear reader, Ctrl+F for “Eilenberg-Mac Lane space” in nCategories and CohomologyIf you are uncomfortable with a group as a category, Ctrl+F for “ordinary particle is a point” in From Loop Space Mechanics to Nonabelian Strings.]

Let $(a,b)$ be a pair of commuting elements in $G$.

In other words, let $(a,b)$ be a pair of paths in $\pi_1(BG)$ that commute. Note that $a$ and $b$ must be loops based at the same point to commute, and that the torus is $S^1 \times S^1$.

We map the first generator of the torus to $a$ and the second generator to $b$.

The first frame of this gif is then $ab$, and the last frame is $ba$ (the middle frames are homotopies).
Any paths in $\pi_1$ that commute (i.e. any pair $(a,b)$ of commuting elements in $G$) give a map of the torus into your space $BG$.