line bundles – good fibrations

The Landweber exact-functor theorem

This post assumes familiarity with formal group laws, the definition of exact sequences, the motivation of the Landweber-Ravenel-Stong construction, that the exactness axioms is one of the generalized Eilenberg-Steenrod axioms, and the fact that formal group laws over $R$ are represented by maps from the Lazard ring to $R$.

Recall the the Landweber-Ravenel-Stong Construction: $MU^*(X) \otimes_{L} R \simeq E^*(X)$, where $MU^* \simeq L$ and $R \simeq E^*(pt)$.

We know that in general, tensoring with abelian groups does not preserve exact sequences (e.g., applying $-\otimes_{\mathbb{Z}} \mathbb{Z}/2$ to $0 \to \mathbb{Z} \xrightarrow{\times p} \mathbb{Z} \to \mathbb{Z}/p \to 0$).

So, when does the functor $-\otimes_L R: MU^*(X) \to E^*(X)$ preserve exact sequences?

Continue reading The Landweber exact-functor theorem

$Pic(X)$ 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 $X$.

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$CP^\infty$ 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