10-06 Notes
Our goal is to understand the cohomology of the space \(\textrm{Map}(\Sigma,BU(n))\) for a closed surface \(\Sigma\). Since we have already shown it to be rationally homotopic to a product of Eilenberg-Maclane spaces, we have two remaining goals:
Show that \(H^{*}(\mathrm{Map}(\Sigma,BU(n));\mathbb{Z})\) is torsion-free.
Identify integral generators of \(H^{*}(\mathrm{Map}(\Sigma,BU(n));\mathbb{Z})\).
The case \(\Sigma=S^{2}\)
Let us suppose for now that \(\Sigma=S^{2}\). For a fixed point \(p\in S^{2}\),
the evaluation map at \(p\) gives a fibration 
whose fibers are homotopic to \(\Omega U(n)\). From Bott’s proof of
Bott periodicity we know that the energy functional on \(\Omega U(n)\)
is a Morse function which only has critical points of even index,
so from Morse theory it follows that \(\Omega U(n)\) has torsion-free
cohomology supported entirely in even dimensions. In fact, this is
still true if we replace \(U(n)\) with any compact Lie group.
Now, we’ve already seen that this fibration is rationally a product in cohomology: \[H^{*}(\mathrm{Map}(S^{2},BU(n));\mathbb{Q})\cong H^{*}(\Omega^{2}BU(n);\mathbb{Q})\otimes_{\mathbb{Q}}H^{*}(BU(n);\mathbb{Q}).\] But both \(\Omega^{2}BU(n)\) and \(BU(n)\) are torsion-free, so if the Serre spectral sequence has any nontrivial differentials over \(\mathbb{Z}\) then their effect should be visible over \(\mathbb{Q}\). Thus the fibration must also be a cohomological product over \(\mathbb{Z}\), or \[H^{*}(\mathrm{Map}(S^{2},BU(n));\mathbb{Z})\cong H^{*}(\Omega^{2}BU(n);\mathbb{Z})\otimes_{\mathbb{Z}}H^{*}(BU(n);\mathbb{Z}).\]
The general case
Let \(\Sigma=\Sigma_{g}\) be the surface of genus \(g\). The quotient
of \(\Sigma\) by its \(1\)-skeleton \(\bigvee_{i=1}^{2g}S^{1}\) is the
sphere \(S^{2}\), giving a fibration 
whose last term is \(\left(\prod^{2g}\mathrm{Map}_{*}(S^{1},BU(n))\right)\times BU(n)=\left(\prod^{2g}U(n)\right)\times BU(n)\),
where \(\mathrm{Map}_{*}\) denotes the space of pointed maps. Once
again the base and fiber are torsion-free in cohomology, and the total
space is rationally their cohomological product, so we can conclude
that it is an integral cohomology product as well.
Remark 1. In dimensions higher than \(2\) this argument can fail if the fiber is not torsion-free. For example, considering \(\mathrm{Map}(S^{4},BU(2))\) we get a fibration with fiber \(\Omega^{4}BU(2)\simeq\Omega^{3}U(2)\). Now topologically we have a homeomorphism \(S^{1}\times SU(2)\cong U(2)\) sending \((e^{i\theta},A)\) to the matrix \(\left(\begin{array}{cc} e^{i\theta}\\ & 1\end{array}\right)A\), and \(SU(2)\cong S^{3}\), so we see that \(\pi_{1}(\Omega^{3}U(2))=\pi_{4}(S^{1}\times S^{3})=\mathbb{Z}/2\mathbb{Z}\).
Generators for \(H^{*}(B\mathcal{G};\mathbb{Z})\)
In order to find generators of \(H^{*}(B\mathcal{G};\mathbb{Z})\), we must find integral generators of \(H^{*}(\Omega^{2}BU(n))\cong H^{*}(\Omega U(n))\). For the infinite unitary group \(U=\underset{n\to\infty}{\lim}U(n)\), we know that \(H^{*}(BU;\mathbb{Z})\) is generated by the Chern classes \(c_{1},c_{2},\dots\) of the universal bundle \(EU\to BU\). Since the inclusion \(U(n)\hookrightarrow U\) induces a map \[\Omega^{2}BU(n)\to\Omega^{2}BU\simeq BU,\] where the homotopy equivalence follows from Bott periodicity, we can hope for a similar construction for \(\Omega^{2}BU(n)\), but this will require a more detailed look at \(BU\).
The space \(BU\) can be modeled as the limit of Grassmanians \(\underset{n\to\infty}{\lim}\mathrm{Gr}_{n}(\mathbb{C}^{2n})\), but it will be more useful to consider another identification:
Theorem 1 (Atiyah). There is a homotopy equivalence \(\mathbb{Z}\times BU\simeq\mathrm{Fred}(H)\), where \(\mathrm{Fred}(H)\) is the space of Fredholm operators on the infinite dimensional separable Hilbert space \(H\) and the \(\mathbb{Z}\) factor captures the indices of these operators.
Proof. First, we note that \(\mathbb{Z}\times BU\) is a classifying space for complex K-theory, i.e. \(K(X)\cong[X,BU]\) where \(K(X)\) is the Grothendieck group of vector bundles over a compact space \(X\). Given a map \(f:X\to\mathrm{Fred}(H)\), and a point \(x_{0}\in X\) with neighborhood \(\mathcal{U}\) small enough so that the orthogonal projection \(\mathrm{Im}(f(x_{0}))^{\perp}\to\mathrm{Im}(f(x))^{\perp}\) is injective for all \(x\in\mathcal{U}\), we can construct a virtual bundle as follows: Choose a finite cover \(\{\mathcal{U}_{i}\}\) of \(X\), together with points \(x_{i}\in\mathcal{U}_{i}\) having this same property, and let \(\{\varphi_{i}\}\) be a partition of unity subordinate to \(\{\mathcal{U}_{i}\}\). We can define a map \[\begin{aligned} \bar{f}:H\oplus\left(\bigoplus_{i=1}^{n}\mathrm{Im}(f(x_{i}))^{\perp}\right) & \to H\\ (h,h_{1},\dots,h_{n}) & \mapsto f(x)\cdot h+\sum_{i=1}^{n}\varphi_{i}(x)\cdot h_{i} \end{aligned}\] which is now surjective, and associate to \(f\) the difference \(\ker(\bar{f})-\oplus_{i=1}^{n}\mathrm{Im}(f(x_{i}))^{\perp}\in K(X)\). We leave it as an exercise to check that this gives a well-defined map \([X,\mathrm{Fred}(H)]\to K(X)\).
Conversely, we need to associate to each virtual bundle \([V]-[W]\in K(X)\) a map \(f:X\to\mathrm{Fred}(H)\). The bundle \([V]-[W]\) can be written equivalently as \([V']-\underline{\mathbb{C}}^{n}\), where \(\underline{\mathbb{C}}\) denotes a trivial line bundle, so we construct a map \(f:V'\oplus H\to\underline{\mathbb{C}}^{n}\oplus H\) defined by \(f(v,h)=(0,h)\). This clearly has kernel \(V'\) and cokernel \(\underline{\mathbb{C}}^{n}\). But by Kuiper’s theorem, which states that \(U(H)\) is contractible, the bundle \(V'\oplus H\) is a trivial bundle over \(H\) and so \(V'\oplus H\cong X\times H\). In particular we can express \(f\) equivalently as a Fredholm operator on \(H\), parametrized by \(x\in X\), as desired. ◻
Having described a model for \(BU\), we would like to explicitly describe an identification \([X,\Omega^{2}BU]\simeq[X,BU]\); note that \([X,\Omega^{2}BU]\simeq[S^{2}\wedge X,BU]\). In fact, there is a natural map \([S^{2}\wedge X,BU]\to[X,BU]\): A given map \(f:S^{2}\wedge X\to BU\) corresponds to a vector bundle \(V-\underline{\mathbb{C}}^{n}\to S^{2}\wedge X\). Constructing a family of \(\bar{\partial}\) operators on the \(S^{2}\) factor at each point, \[\bar{\partial}:\Omega^{0,0}(V|_{S^{2}\times\{x\}})\to\Omega^{0,1}(V|_{S^{2}\times\{x\}}),\] gives us the desired map \(X\to\mathrm{Fred}(H)\):
Theorem 2. This construction gives an isomorphism \[[X,\Omega^{2}BU]\simeq[S^{2}\wedge X,BU]\stackrel{\sim}{\longrightarrow}[X,BU].\]
The evaluation map \(\mathrm{ev}:S^{2}\times\mathrm{Map}_{*}(S^{2},BU(n))\to BU(n)\)
pulls back to a bundle \(V=\mathrm{ev}^{*}(EU(n))\): 
and if we identify \(V\) with the corresponding \(\sigma\in K(\mathrm{Map}_{*}(S^{2},BU(n)))\),
the Chern classes \(c_{i}(\sigma)\) will be integral generators of
\(H^{*}(\Omega^{2}BU(n))\) as desired.
Similarly, given a vector bundle \(E\to\Sigma\) and a connection \(A\in\mathcal{A}\) on \(\Sigma\), we have a family of Fredholm operators \(\mathcal{A}\to\mathrm{Fred}(H)\) of the form \[A\mapsto(\bar{\partial}_{A}:\Omega^{0,0}(E)\to\Omega^{0,1}(E)).\] This family is \(\mathcal{G}\)-equivariant in the sense that \(\bar{\partial}_{g\cdot A}=g^{-1}\circ\bar{\partial}_{A}\circ g\), so it corresponds to an element of the equivariant K-theory \(K_{\mathcal{G}}(\mathcal{A})\), whose Chern classes \(c_{i}\) are generators of \(H_{\mathcal{G}}^{2i}(\mathcal{A})\cong H^{2i}(B\mathcal{G})\).
Minima of the Yang-Mills functional
Ideally, we would like the minima of the Yang-Mills functional to be free orbits of \(\mathcal{G}\). Recall that in general, if \(d_{A}*F_{A}=0\) then we can write \(E\) as a sum \(\bigoplus_{\lambda}E_{\lambda}\), where up to a constant factor the \(\lambda\) are the eigenvalues of \(*F_{A}\) and we have \(\lambda=\frac{c_{1}(E_{\lambda})}{\mathrm{rk}(E_{\lambda})}\). The Yang-Mills functional in this case satisfies (again up to a constant) \[\mathcal{E}(A)=\sum_{\lambda}\left(\frac{c_{1}(E_{\lambda})}{\mathrm{rk}(E_{\lambda})}\right)^{2}\cdot\mathrm{rk}(E_{\lambda})=\sum_{\lambda}\frac{c_{1}(E_{\lambda})^{2}}{\mathrm{rk}(E_{\lambda})},\] and so to minimize \(\mathcal{E}(A)\) when \(n=c_{1}(E)=\sum_{\lambda}c_{1}(E_{\lambda})\) is fixed, we need each of the \(c_{1}(E_{\lambda})\) to be as close to equal as possible.
Example 1. Consider \(U(2)\) bundles over \(\Sigma\). Either \(E=E_{\lambda}\) does not split, so \(\mathcal{E}(A)=\frac{n^{2}}{2}\), or \(E=L_{1}\oplus L_{2}\) is a sum of line bundles with Chern classes \(n_{1}\) and \(n_{2}\) and then \(\mathcal{E}(A)=n_{1}^{2}+n_{2}^{2}\). If \(n\) is odd, then \(n_{1}\not=n_{2}\) and so \(n_{1}^{2}+n_{2}^{2}>\frac{n^{2}}{2}\), meaning that we cannot have minima which are direct sums of line bundles. On the other hand, if \(n\) is even and \(|n_{1}|=|n_{2}|=\frac{|n|}{2}\) then minima can be direct sums. This means that when \(n\) is odd, the minima \(A\) all satisfy \(\mathrm{Stab}(A)=U(1)\), consisting of constant gauge transforms, but for \(n\) even the stabilizer can jump from \(U(1)\) to \(U(1)\times U(1)\) among the minimal action solutions.
In the next lecture, we’ll continue this analysis for higher rank bundles and see that the analogous condition is for \(c_{1}(E)\) and \(\mathrm{rk}(E)\) to be coprime.
