Course website: https://akramalishahi.github.io/CharClass.html

Description from Akram’s syllabus:

This course is about characteristic classes, which are cohomology classes naturally associated to vector bundles or, more generally, principal bundles. They are a key tool in modern {algebraic, differential}×{topology, geometry}. The course starts with an introduction to vector bundles and principal bundles. It then discusses their main characteristic classes—the Euler class, Stiefel-Whitney classes, Chern classes, and Pontryagin classes. The last part of the class discusses some applications of characteristic classes to bordisms. In the process, we will see some nice applications (e.g., to immersions) and review some important parts of algebraic topology (e.g., obstruction theory).

References:

• [Hu?] Husemoller, Fiber bundles.
• [MS?] Milnor and Stasheff, Characteristic classes.
• [S?] Steenrod, The topology of fibre bundles.
• [Ha?] Hatcher, Vector bundles and K-theory .
• [BottTu?] Bott and Tu, Differential forms in algebraic topology.

Prerequisites:

• Smooth manifolds: smooth maps and derivatives, differential forms.
• Algebraic topology: homology and cohomology.

An overview of what we’ll cover:

• General definitions and constructions related to vector bundles and fiber bundles.

  Why bundles? For a bundle \(E \xrightarrow{\pi} B\), characteristic classes will be cohomology classes in \(H^*(B)\). Natural examples include

  • The tangent bundle \(TX\to X\), and vector fields will be sections.
  • Exterior products \(\bigwedge\nolimits^n TX\), where differential forms live
  • Normal bundles \(\nu\), giving directions an embedded submanifold can be deformed.

  Also note that manifolds locally look like vectors spaces (\({\mathbb{R}}^n\)!) and so embedded manifolds locally look like vector bundles. In particular, if \(f: M^n \hookrightarrow N^k\) is an embedding, locally \(\nu\) is locally a \(k-n\) dimensional vector bundle over \({\mathbb{R}}^n\) (and globally a bundle of the form \(\nu: E \to f(M_n)\))

  

• Characteristic Classes: Euler, Stiefel-Whitney, Pontryagin, etc.

  These package geometric information into algebraic invariants that are often computable. Some examples:

  • Stiefel-Whitney classes can detect if \(M^n = {{\partial}}M^{n+1}\) is a boundary (for smooth closed manifolds).
  • Euler classes can prove the Hairy Ball theorem, i.e. \(S^2\) admits no nonvanishing continuous vector fields, which can be generalized to \(S^{2n}\) and to splitting the tangent bundle.
  • Pontryagin classes: Milnor used these to produce exotic \(S^7\)s! These are manifolds \(M^7\) which are homeomorphic but not diffeomorphic to \(S^7\).
  • Chern classes.

Note that this necessarily implies that all fibers are homeomorphic, noting that \(F_b \mathrel{\vcenter{:}}=\pi^{-1}(b) \xrightarrow{\phi} \left\{{b}\right\} \times F\). We have inclusions: vector bundles \(\implies\) fiber bundles \(\implies\) fibrations. For a fibration that’s not a fiber bundle, one can collapse a fiber in a trivial bundle, e.g. 

Consider the following setup:

• \(B\in {\mathsf{Top}}\)
• \(\pi:E\to B\) is a map of underlying sets
• There is a bundle atlas \(\left\{{\phi_\alpha}\right\}\), each \(\phi_\alpha\) being a bijection.

Then there exists at most one topology on \(E\) such that \(\pi: E\to B\) is a fiber bundle with the given atlas.

We have some natural operations:

1. Direct sums.

For \(E_1, E_2 \in { { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }}_{/ {B}}\), so \(E_1 \xrightarrow{\pi_1} B\) and \(E_2 \xrightarrow{\pi_2} B\), we can form \(E_1 \oplus E_2 \xrightarrow{\pi} B\). As a set, take \begin{align*} E_1 \oplus E_2 \mathrel{\vcenter{:}}=\displaystyle\bigcup_{b\in B} F_{1, b} \oplus F_{2, b} \end{align*} as a union of direct sums of vector spaces. For the bundle map, take \(\pi(F_{1, b} \oplus F_{2, b}) \mathrel{\vcenter{:}}=\left\{{b}\right\}\). For charts, for any \(b\in B\) pick individual charts about \(b\), say \((U_1, \phi_1)\) for \(E_1\) and \((U_1, \phi_2)\) for \(E_2\), form charts \begin{align*} \left\{{ (U_1 \cap U_2, \phi: \pi^{-1}(U_1 \cap U_2) \to {\mathbb{R}}^{n_1 + n_2})}\right\} \end{align*} where \(n_1 \mathrel{\vcenter{:}}=\dim_{\mathbb{R}}F_{1, b}\) and \(n_2 \mathrel{\vcenter{:}}=\dim_{\mathbb{R}}F_{2, b}\) and define \((b, (v_1, v_2)) \xrightarrow{\phi} (\phi_1(v_1), \phi_2(v_2))\).

Setup:

• \(B \in {\mathsf{Top}}\) is a space.
• \(\pi:E\to B\) is a map of sets with fibers/preimages \(F \mathrel{\vcenter{:}}= F_b \mathrel{\vcenter{:}}=\pi^{-1}(\left\{{b}\right\})\).
• A bundle atlas for \(\pi\) is \(\phi\) where \(\phi_U: \pi^{-1}(U) \to U \times F\) is a bijection of sets

Then there is at most one topology on \(E\) making \(\pi:E\to B\) into a fiber bundle with the specified atlas.

Consider what happens on overlapping charts – looking at maps fiberwise yields:

Link to Diagram

Starting at \((U \cap V)\times F\) and running the diagram clockwise yields a map \begin{align*} \phi_V \circ \phi_U ^{-1}: (U \cap V ) \times F &\to (U \cap V ) \times F \\ (b, f) &\mapsto (b, \phi_{VU}(f) ) \end{align*} where \(\phi_{VU}\) is the following continuous map, defining a transition function: \begin{align*} \phi_{VU}: U \cap V &\to {\operatorname{Homeo}}(F) ,\end{align*} where \({\operatorname{Homeo}}(F) \mathrel{\vcenter{:}}=\mathop{\mathrm{Hom}}_{{\mathsf{Top}}}(F, F)\) with the compact-open topology.

Today: relating \(\mathop{\mathrm{Prin}}{\mathsf{Bun}}_{/ {G}}\) to fiber bundles with a \(G{\hbox{-}}\)structure. Recall that a principal \(G{\hbox{-}}\)bundle is a fiber bundle \(\pi:P\to B\) with a fiberwise \(G{\hbox{-}}\)action \(P\times G\to P\) which induces a free and transitive action on each fiber. Note that we assume \(G\in{\mathsf{Top}}{\mathsf{Grp}}\). Any bundle in \(\mathop{\mathrm{Prin}}{\mathsf{Bun}}_{/ {G}}\) is a fiber bundle with fibers \(F\) homeomorphic to \(G\) and admits a \(G{\hbox{-}}\)structure: \begin{align*} G &\hookrightarrow{\operatorname{Homeo}}(G) \\ g &\mapsto (h\mapsto gh) .\end{align*} Using that \(F\cong G\), taking charts \((U, \varphi), (V, \psi)\) for \(\pi:P\to B\), we can identify

Link to Diagram

So every transition function is given by left-multiplication by some element in \(G\), as opposed to arbitrary homeomorphisms of \(G\) as a topological group.

Given a principal \(G{\hbox{-}}\)bundle \(\pi:P\to B\) and a \(F\in {\mathsf{Top}}\) with a left \(G{\hbox{-}}\)action. Then define \begin{align*} P { \underset{\scriptscriptstyle {G} }{\times} } F / (pg, f)\sim (p, gf) \end{align*} as a fiber bundle over \(B\) using \(\pi\) as the projection. Note that this looks like a tensor product, and this works in general for any space \(P\) with a right \(G{\hbox{-}}\)action and \(F\) with a left \(G{\hbox{-}}\)action. This will be a fiber bundle with fiber \(F\) and structure group \(G \leq {\operatorname{Homeo}}(F)\).

Locally there is a homeomorphism: \begin{align*} (U\times G) \overset{\scriptscriptstyle {G} }{\times} F &\xrightarrow{\sim} U\times F \\ (p, g, f) &\mapsto (p, gf) .\end{align*} This is well defined since \((p, gh, f)\) and \((p, g, hf)\) map to \((p, ghf)\). The inverse is \((p, f) \mapsto (p, 1, gf)\).

We want to show the equivalence between (isomorphism classes) of fiber bundles with \(G\) structures with fiber \(F\) and principal \(G{\hbox{-}}\)bundles. Recall that \(\mathop{\mathrm{Prin}}{\mathsf{Bun}}_{/ {G}}\) are fiber bundles \(P \xrightarrow{\pi} B\) with a right fiberwise \(G{\hbox{-}}\)action which is free and transitive on each fiber.

To send fiber bundles to principal bundles, we used a mixing construction. Since \(G\curvearrowright F\), we get an identification \(G \subseteq {\operatorname{Homeo}}(F, F)\). We constructed \begin{align*} P { \underset{\scriptscriptstyle {G} }{\times} } F \mathrel{\vcenter{:}}=(P\times F)/(pg, f)\sim (p, gf) .\end{align*} A lemma was that \(P{ \underset{\scriptscriptstyle {G} }{\times} } F \to B\) is a fiber bundle with fiber \(F\) and projection \(\pi(p, f) \mathrel{\vcenter{:}}=\pi(p)\).

Today we’ll talk about the reverse direction. Note the composition of sending \(E\) to \(\mathop{\mathrm{Prin}}{\mathsf{Bun}}_{/ {G}}\) and then mixing recovers \(E\) when \(E\) is a vector bundle, but not generally. \begin{align*} \left\{{\substack{ \text{Fiber bundles with }G{\hbox{-}}\text{structures} \\ \text{ and fiber }F }}\right\} \adjunction{\text{Clutching}}{\text{Mixing}}{}{} \left\{{\substack{ \text{Principal $G{\hbox{-}}$bundles} }}\right\} \end{align*}

Note that if we choose a basis for the fibers, we can set \(A' \mathrel{\vcenter{:}}={\left[ {\mathbf{e}_1, \cdots, \mathbf{e}_n} \right]}^t\) to be the matrix with columns \(\mathbf{e}_i\), the map \(f\) is given by \(f(A', \mathbf{b}) \mathrel{\vcenter{:}}= A'\mathbf{b}\), and we’re showing that \((A'A)\mathbf{b} = A'(A\mathbf{b})\). However, this involves choosing an isomorphism between the abstract fibers and \({\mathbb{R}}^n\).

What are local charts for a principal bundle? For \(P{ \underset{\scriptscriptstyle {G} }{\times} } F\), pick charts \((U, \phi)\) for \(P \xrightarrow{\pi} B\): \begin{align*} \phi: \pi^{-1}(U) &\to U \times G \\ x & \mapsto (\pi(x), \gamma(x)) .\end{align*} Then a local chart for the principal bundle is of the form \begin{align*} \pi^{-1}(U) { \underset{\scriptscriptstyle {G} }{\times} } F &\xrightarrow{\tilde \phi} U \times F \\ (x, f) &\mapsto ( \pi(x), \gamma(x) f) .\end{align*} We also have \begin{align*} (U \times G){ \underset{\scriptscriptstyle {G} }{\times} } F &\to U \times F \\ ( (x, g), f) &\mapsto (x, gf) .\end{align*} One can invert \(\tilde \phi\) using \((a, f) \mapsto (\phi^{-1}(a, 1), f)\). This yields transition functions: writing \begin{align*} \phi_V: \pi^{-1}(V) &\to V \times G \\ x &\mapsto (\pi(x), \psi(x) ) ,\end{align*} then \begin{align*} \phi_{VU} = \phi_V \circ \phi_U^{-1}: (a, f) & \\ &\xrightarrow{\phi_U^{-1}} ( \phi^{-1}_U(a, 1), \,\, f) \\ &\xrightarrow{\phi_V} (\pi \phi_U^{-1}(a, 1), \,\, \psi( \phi_U^{-1}(a, 1))f ) \\ &= (a,\,\, \psi(\phi_U^{-1}(a, 1)) f) .\end{align*} This says that \((a, 1) \mapsto \psi( \phi^{-1}_U(a, 1))\).

In general, for a bundle \(E \xrightarrow{\pi} B\), taking local trivializations \(\phi_U, \phi_V\), we get \(\phi_{VU}: (U \cap V ) \times F {\circlearrowleft}\), or currying an argument, \(\phi_{VU}: U \cap V \to {\operatorname{Homeo}}(F, F)\). If the bundle satisfies the cocycle condition \(\phi_{UW} = \phi_{VW} \circ \phi_{UV}\). Given a covering \(\left\{{U_i}\right\}_{i\in I}\rightrightarrows B\), we get \(\phi_{ij}: U_i \cap U_j \to G\) and a topological space \(F\) with \(G \subseteq {\operatorname{Homeo}}(F, F)\) satisfying the cocycle condition \(\phi_{ik} = \phi_{jk} \circ \phi_{ij}\), then we can build a fiber bundle with fiber \(F\) and structure group \(G\) by setting \(E = {\textstyle\coprod}_{i\in I} (U_i \times F)/\sim\) We then set for \(b\in U_i \cap U_j\) the equivalence \begin{align*} (U_i \times F) \ni (b, f)\sim (b, \phi_{ij}(b) f) \in (U_j \times F) .\end{align*} This is an equivalence relation precisely when the cocycle condition holds. This is referred to as clutching data.

Recall that we have a correspondence \begin{align*} \left\{{\substack{ \text{Vector bundles }E }}\right\} &\adjunction{\text{clutching}}{\text{mixing}}{}{} \left\{{\substack{ \text{Principal $\operatorname{GL}_n{\hbox{-}}$bundles $\mathop{\mathrm{Frame}}(E)$} }}\right\} \end{align*} We saw that \(E \cong \mathop{\mathrm{Frame}}(E) { \underset{\scriptscriptstyle {\operatorname{GL}_n({\mathbb{R}})} }{\times} } {\mathbb{R}}^n\). If we take \(\mathop{\mathrm{Frame}}(E)\), mix, and apply the clutching construction, is the result bundle-isomorphic to the frame bundle?

Recall the clutching construction: we take a cover \(\left\{{U_i}\right\}_{i\in I}\) and \(\phi_{ij}: U_i \cap U_j \to G\) satisfying the cocycle condition \(\phi_{ij}\phi_{jk} = \phi_{ik}\), then \(G \subseteq {\operatorname{Homeo}}(F, F)\) and we construct a fiber bundle \(\displaystyle\bigcup_{i\in I} U_i \times F / \sim\) where for \(b\in (U_i \cap U_j )\) and \begin{align*} (b, f) \in (U_i \cap U_j )\times F \subseteq U_i\times F ,\end{align*} we send this to \begin{align*} (b, \phi_{ji}(b) f ) \in (U_i \cap U_j)\times F \subseteq U_j \times F .\end{align*} This will be a fiber bundle with fiber \(F\) and structure group \(G\). Moreover, if \(F=G\), this will be a principal \(G{\hbox{-}}\)bundle using right-multiplication.

For principal bundles, these \(\gamma_i\) will give sections assembling to a global section obtained from clutching data:

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The map on \(U_i \to \displaystyle\bigcup_i U_i \times F\) will be \((b, \gamma_i(b))\), and we can use that \begin{align*} (b, \gamma_i(b)) \sim (b, \phi_{ji}(b)\gamma_i(b) ) \sim (b, \gamma_j(b)) ,\end{align*} so these agree on overlaps.

One can check that \(\delta^1 \circ \delta^0 = 0\) is trivial. And 1-cocycle will yield a fiber bundle.

This works when bundles have the same open cover, and if not, we can take a common refinement.

Recall that given a \(B\in {\mathsf{Top}}\) and \({\mathcal{U}}\rightrightarrows B\), we defined \({\check{H}}_1({\mathcal{U}}; G)\) which classified isomorphism classes of fiber bundles \(E \xrightarrow{\pi} B\) with fiber \(F\), \(G \subseteq {\operatorname{Homeo}}(F)\), and structure group \(G\), given by clutching data using \({\mathcal{U}}\). The cochains were given by the following:

\begin{align*} {\check{C}}^0({\mathcal{U}}; G) &= \left\{{\left\{{ \gamma_i: U_i \to G}\right\}_{i\in I}}\right\}\\ {\check{C}}^1({\mathcal{U}}; G) &= \left\{{\left\{{ \phi_{ij}: U_i \cap U_j \to G }\right\}_{i, j \in I}}\right\}\\ {\check{C}}^2({\mathcal{U}}; G) &= \left\{{\left\{{ \eta_{ijk}: U_i \cap U_j \cap U_k \to G }\right\}_{i,j,k\in I}}\right\} \end{align*} with boundary maps \(\delta_i: {\check{C}}^{i-1} \to {\check{C}}^i\): \begin{align*} (\delta_1 \gamma)_{ij} &= \gamma_i \gamma_j^{-1}\\ (\delta_2 \phi)_{ijk} &= \phi_{ij} \phi_{jk} \phi_{ik}^{-1} .\end{align*} Note that

• \(\delta_2 \circ \delta_1 = 0\)
• \(\ker \delta_2 = Z^1({\mathcal{U}}; G)\) yields clutching data, i.e. a fiber bundle with fiber \(F\),
• \(\operatorname{im}\delta_1\) yields trivial bundles,
• \({\check{H}}^1({\mathcal{U}}; G) \mathrel{\vcenter{:}}= Z^1({\mathcal{U}}; G) / \operatorname{im}({\check{C}}^0({\mathcal{U}}; G) \to Z^1({\mathcal{U}}; G))\).

We’ll see that \((\gamma \varphi)_{ij} = \gamma_i \varphi_{ij} \gamma_j^{-1}\), and by a lemma this will prove the above claim about classifying isomorphism classes.

Since any two covers have a common refinement, we’ll assume \({\mathcal{V}}\leq {\mathcal{U}}\) is always a refinement. We can then restrict clutching data from \({\mathcal{U}}\) to \({\mathcal{V}}\): given \(\left\{{\phi_{ij}}\right\}_{i,j\in I}\), we can set \(\psi_{ij} \mathrel{\vcenter{:}}={ \left.{{ \phi_{f(i), f(j)}}} \right|_{{V_i \cap V_j}} }\), noting that if \(V_j \subseteq U_{f(j)}\) and \(V_i \subseteq U_{f(i)}\) then \(V_i \cap V_j \subseteq U_{f(i)} \cap U_{f(j)}\). These yield maps \(\psi_{ij}: V_i \cap V_j \to G\) satisfying the cocycle condition, so \(\psi_{ij} \in Z^1({\mathcal{V}}; G)\). This means that we have map \(Z^1({\mathcal{U}}; G)\to Z^1({\mathcal{V}}; G)\) which respects the actions of \({\check{C}}^0({\mathcal{U}}; G), {\check{C}}^0({\mathcal{V}}; G)\) respectively. Since the category of covers with morphisms given by refinements come from a preorder, we can take a colimit to define \begin{align*} {\check{H}}^1(B; G) \mathrel{\vcenter{:}}=\colim_{{\mathcal{U}}\rightrightarrows B} {\check{H}}^1({\mathcal{U}}; G) .\end{align*}

Note that we need to impose the cocycle condition, since lifts may not be unique and some choices may not glue correctly!

So the functor \(X\mapsto \mathop{\mathrm{Prin}}_G(X)\) is contravariant functor.

Last time: the bundle homotopy lemma. If \(P\to I\times X \in \mathop{\mathrm{Prin}}{\mathsf{Bun}}(G)\), then there is a bundle isomorphism

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where \({ \left.{{f}} \right|_{{0\times P_0}} }\) is the identity.

We looked at a theorem stating the correspondence \begin{align*} \mathop{\mathrm{Prin}}{ {\mathsf{Bun}}\qty{G} }_{/ {X}} \rightleftharpoons[X, {\mathbf{B}}G] .\end{align*}

The same proof shows the following:

Uniqueness up to homotopy:

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Then since \((h'h^{-1})^* E \cong E\), \(h' h^{-1}\simeq\operatorname{id}\) and \(h (h')^{-1}\simeq\operatorname{id}\), so we get a homotopy equivalence \(B \simeq B'\).

We’ll prove this theorem using Segal’s construction. For discrete groups \(G\), the construction is covered in Hatcher 1B. Hatcher constructs \(K(G, 1)\), a space with \(\pi_1 = G\) and contractible universal cover. Then the universal cover \(\widehat{X}\to X\) is a principle \(G{\hbox{-}}\)bundle, and since \(\widehat{X}\) is contractible, it is universal:

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Let \(G\in {\mathsf{Grp}}\), and consider the following two categories. \({\mathbf{B}}G\) will be the category:

• \({\operatorname{Ob}}({\mathbf{B}}G) = \left\{{{\operatorname{pt}}}\right\}\).
• \(\mathop{\mathrm{Mor}}_{{\mathbf{B}}G}({\operatorname{pt}}, {\operatorname{pt}}) = \left\{{g\in G}\right\}\), i.e. there is one morphism for every group element, with composition \(g_1 \circ g_2 \mathrel{\vcenter{:}}= g_1g_2\) given by group multiplication.

\(E G\) will be the category:

• \({\operatorname{Ob}}(EG) = \left\{{g\in G}\right\}\), one object for each element of \(G\),
• \(\mathop{\mathrm{Mor}}(g, h)=\left\{{g^{-1}h}\right\}\), a single (conveniently labeled!) morphism for each ordered pair \((g, h)\).

Note that \(G\curvearrowright EG\):

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This induces an action on \({ \mathcal{N}({EG}) } \in {\mathsf{Top}}\), where the 0-simplices correspond to elements of \(G\). and \(n{\hbox{-}}\)simplices are chains \begin{align*} g_0 \xrightarrow{g_0^{-1}g_1} g_1 \xrightarrow{g_1^{-1}g_2} g_2 \to \cdots \to g_n .\end{align*} Acting on this by \(G\) yields \begin{align*} gg_0 \xrightarrow{g_0^{-1}g_1} gg_1 \xrightarrow{g_1^{-1}g_2} gg_2 \to \cdots \to gg_n ,\end{align*} noting we leave the morphism labeling unchanged, and that uniqueness of morphisms makes the simplicial boundary map behave nicely.

Note that \begin{align*} { \mathcal{N}({{\mathbf{B}}G}) } &= \Delta^0 {\textstyle\coprod}\Delta^1 \times G {\textstyle\coprod}\Delta^2 \times G{ {}^{ \scriptscriptstyle\times^{2} } } {\textstyle\coprod}\Delta^3 \times G{ {}^{ \scriptscriptstyle\times^{3} } }\cdots \\ { \mathcal{N}({E G}) } &= \Delta^0 \times G {\textstyle\coprod}\Delta^1 \times G{ {}^{ \scriptscriptstyle\times^{2} } } {\textstyle\coprod}\Delta^2 \times G{ {}^{ \scriptscriptstyle\times^{3} } }\cdots ,\end{align*} where the gluing data for \({ \mathcal{N}({{\mathbf{B}}G}) }\) is given by \begin{align*} {{\partial}}_n: \Delta^n \times G{ {}^{ \scriptscriptstyle\times^{n} } } &\to \Delta^{n-1} \times G{ {}^{ \scriptscriptstyle\times^{n-1} } } \\ (\mathbf{x}, \mathbf{g}) &\mapsto (\mathbf{x}\setminus\left\{{x_n}\right\}, \mathbf{g} \setminus\left\{{g_n}\right\}) \end{align*} and for \({ \mathcal{N}({EG}) }\) is \begin{align*} {{\partial}}_n: \Delta^n \times G{ {}^{ \scriptscriptstyle\times^{n+1} } } &\to \Delta \times G{ {}^{ \scriptscriptstyle\times^{n} } } \\ (\mathbf{x}, \mathbf{g}) &\mapsto (\mathbf{x}\setminus\left\{{x_n}\right\}, \mathbf{g} \setminus\left\{{g_n}\right\}) .\end{align*}

The action \(G\curvearrowright EG\) is the following: \begin{align*} g\cdot (\mathbf{x}, \mathbf{g}) \mapsto (\mathbf{x}, {\left[ {gg_0, gg_1,\cdots, gg_n} \right]} ) .\end{align*}

This clearly holds for \(E G\), since every object is initial and terminal.

Today: a short discussion on generalizations of \({\mathbf{B}}G\) to topological groups.

We can take nerves of topological categories; this just requires tracking the additional enrichment (i.e. the various topologies). The same proof will yield a principal \(G{\hbox{-}}\)bundle \({ \mathcal{N}({{\mathcal{E}}G}) } \xrightarrow{\pi} { \mathcal{N}({{\mathcal{B}}G}) }\), noting that \(G\) again acts on \({ \mathcal{N}({{\mathcal{E}}G}) }\).

We can’t necessarily extend over the entire \(m+1\) skeleton! But here extending it over a retractable neighborhood was enough, so we needed \(G\) to be an ANR in order for \({{\mathbf{B}}G}\) to be an ANR. Why: consider \begin{align*} p^{-1}(U_m) \subseteq \displaystyle\bigcup_{i\leq m} \Delta^i \times G{ {}^{ \scriptscriptstyle\times^{i} } } \subseteq \displaystyle\bigcup_{i\leq m-1} \Delta^i \times G{ {}^{ \scriptscriptstyle\times^{i} } } .\end{align*} If \(G\) is an ANR, use that \(\Delta^i\) is an ANR and so their product will be, then pick a neighborhood and apply \(p\) to get the required open.

We’ll assume all spaces paracompact from this point forward! We have a correspondence \begin{align*} \left\{{\substack{ \text{$n{\hbox{-}}$dimensional CW complexes } }}\right\} \rightleftharpoons \left\{{\substack{ \text{$n{\hbox{-}}$dimensional vector bundles } \\ \text{with an ${\operatorname{O}}_n{\hbox{-}}$structure} }}\right\} \rightleftharpoons \mathop{\mathrm{Prin}}{\mathsf{Bun}}({\operatorname{O}}_n)_{/ {X}} \rightleftharpoons [X, {\mathbf{B}}{\operatorname{O}}_n] \end{align*} Our next goal is to construct \({{\mathbf{B}}{\operatorname{O}}}_n\) and \({\mathsf{E} {\operatorname{O}}}_n\) as spaces. Let \(V_n({\mathbb{R}}^k) \mathrel{\vcenter{:}}=\left\{{(\mathbf{v}_1, \cdots, \mathbf{v}_n) \text{orthonormal} }\right\}\). Note that \({\operatorname{O}}_n\curvearrowright V_n({\mathbb{R}}^k)\) by \begin{align*} \qty{\mathbf{v}_1, \cdots, \mathbf{v}_n} \cdot A = \qty{\sum_i a_{i, 1} \mathbf{v}_i, \sum_i a_{i, 2} \mathbf{v}_i, \cdots, \sum_i a_{i, n} \mathbf{v}_i} .\end{align*}

There is a projection

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We’ll use the fact that \(V_1({\mathbb{R}}^k)\) is \((k-2){\hbox{-}}\)connected, since it’s homotopy equivalent to \(S^{k-1}\).

Using the inclusions \(V_n({\mathbb{R}}^k) \hookrightarrow V_n({\mathbb{R}}^{k+1})\), we can define \(V_n({\mathbb{R}}^\infty) = \colim_k V_n({\mathbb{R}}^k) = \displaystyle\bigcup_{k\geq 0}V_n ({\mathbb{R}}^k)\). We equip it with the weak topology, i.e. \(U \subseteq V_n({\mathbb{R}}^\infty)\) is open iff \(U \cap V_n({\mathbb{R}}^k)\) is open for all \(k\).

It will turn out that \({\mathsf{E} {\operatorname{O}}}_n = V_n({\mathbb{R}}^\infty)\), sometimes referred to as the Stiefel manifold of \(n{\hbox{-}}\)frames in \({\mathbb{R}}^\infty\).

Last time: we were trying to construct \({\mathsf{E} {\operatorname{O}}}_n\) and \({{\mathbf{B}}{\operatorname{O}}}_n\), and we defined \(V_n({\mathbb{R}}^\infty) = \directlim_{k} V_n({\mathbb{R}}^k)\), where \(V_n({\mathbb{R}}^k)\) was the space of \(n\) orthonormal vectors in \({\mathbb{R}}^k\). There is a map \(V_n({\mathbb{R}}^\infty)\to {\operatorname{Gr}}_n({\mathbb{R}}^\infty)\), which will be our candidate for \({\mathsf{E} {\operatorname{O}}}_n \to {{\mathbf{B}}{\operatorname{O}}}_n\).

• Replace \({\operatorname{O}}_n\) with \(U_n\) and \({\mathbb{R}}\) with \({\mathbb{C}}\) to get \({\operatorname{Gr}}_n({\mathbb{C}}^\infty) = {\mathbf{B}}U_n\).
• \(V_n({\mathbb{R}}^\infty)/ {\operatorname{SO}}_n = {\mathbf{B}}{\operatorname{SO}}_n\) yields the Grassmannian of oriented planes.
• For \(H\leq G\), we have \({\mathbf{E}}H = {\mathbf{E}}G\) and \({\mathbf{B}}H = {\mathbf{E}}G/H\).

An alternative description of \({\mathsf{E} {\operatorname{O}}}_n\) and \({{\mathbf{B}}{\operatorname{O}}}_n\), due to Milnor-Stasheff: write \({{\mathbf{B}}{\operatorname{O}}}_n = {\operatorname{Gr}}_n({\mathbb{R}}^\infty)\) and define the canonical bundle \(\gamma\). Recall that every principal \({\operatorname{O}}_n\) bundle is a pullback of the following form:

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Moreover, \(\mathop{\mathrm{Prin}}({\operatorname{O}}_n)_{/ {X}} = [X, {{\mathbf{B}}{\operatorname{O}}}_n] = [X, {\operatorname{Gr}}_n({\mathbb{R}}^\infty)]\). Then \(\gamma_n \xrightarrow{\pi} {\operatorname{Gr}}_n({\mathbb{R}}^\infty)\) is the \({\mathbb{R}}^n{\hbox{-}}\)bundle where \(\pi^{-1}(v) = v = V\), regarded as a plane in \({\mathbb{R}}^\infty\). Another description comes from the mixing construction: \(\gamma_n = V_n({\mathbb{R}}^\infty) \overset{\scriptscriptstyle {{\operatorname{O}}_n} }{\times} {\mathbb{R}}^n \to {\operatorname{Gr}}_n({\mathbb{R}}^\infty)\).

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We proved surjectivity last time for \(X \in {\mathsf{CW}}\) compact, using compactness to embed any bundle into a trivial bundle. We proved injectivity in the form of \(f^* \gamma_n \cong g^* \gamma_n \implies f\simeq g\), again for \(X\in {\mathsf{CW}}\) compact. So we need to handle the case of \(X\) not compact.

Idea: write \(X = \colim_n X^{(n)}\) as a limit of compact finite skeleta, define maps \(f_n: X^{(n)} \to {{\mathbf{B}}{\operatorname{O}}}_n\) and \(f_{n+1}: X^{(n+1)}\to {{\mathbf{B}}{\operatorname{O}}}_n\), then modify \(\tilde f_{n+1} \simeq f_{n+1}\) to extend \(f_n\) in such a way that \(f_n^* \gamma_i = { \left.{{E}} \right|_{{B_n}} }\).

We can write \(F(B) = [B, B] \ni \operatorname{id}_B\), and it turns out that the characteristic class is determined by where \(\operatorname{id}_B\) is sent:

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For us, taking \(B\mathrel{\vcenter{:}}={\operatorname{Gr}}_n({\mathbb{R}}^\infty)\) and \(G(B) = H^j({\operatorname{Gr}}_n({\mathbb{R}}^\infty)) \ni \alpha = c(\operatorname{id}_{B})\), so we can pullback to define \(c(f) = f^*\alpha \in H^j(X)\).

Last time: defining characteristic classes. Recall that given \(F, G \in {\mathsf{Fun}}({\mathsf{Top}}, {-})\), a characteristic class with values in \(G\) is a natural transformation \(c: F { { \, \xrightarrow{\sim}\, }}G\). If \(F\) is representable, then characteristic classes \(c: F\to G\) is of the form \(c(\operatorname{id}_B) \in G(B)\) for \(\operatorname{id}_B \in [B, B] \cong F(B)\), since \(c\) is determined by where it sends \(\operatorname{id}_B\). Recall that Eilenberg-MacLane spaces \(K(G, n)\) represent \(H^n({-}; G)\) for \(G\in {\mathsf{Ab}}{\mathsf{Grp}}\), and are characterized by \(\pi_i(K(G, n)) = G\) only in degree \(i=n\).

We can define \(c_1\) for vector bundles of any dimension by taking a top exterior power to get a line bundle:

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So \begin{align*} c_1(E) \mathrel{\vcenter{:}}= c_1(\bigwedge\nolimits^n E) .\end{align*}

There is a natural isomorphism \(c(f^*(E)) \cong f^*(c(E))\), since we can take iterated pullbacks:

So we can identify \(c(f^*(E)) = ( g\circ f)^* \alpha\) and \(f^*(c(E)) = f^*(g^* \alpha) = (g\circ f)^* \alpha\).

Note that any vector bundle with a Riemannian metric admits a unit disc bundle.

Given a bundle \(E\to X\) and taking its disc bundle \({\mathbb{D}}E\to X\), taking boundaries on fibers yields a sphere bundle \({\mathbb{S}}E\to X\), so \({\mathbb{S}}E_b \mathrel{\vcenter{:}}={{\partial}}{\mathbb{D}}E_b\) on fibers. Note the \({\mathbb{D}}E \simeq X\) by a deformation retraction.

Recall that we were proving the Thom isomorphism theorem. Some motivation:

Without the oriented assumption, this theorem still holds with \(C_2\) coefficients.

Note also that we have naturality for characteristic classes: given a pullback

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Then \begin{align*} c(f^* E) = f^*c(E) .\end{align*} Note that we can always pull back the canonical:

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If the Euler class \(e\) is natural, then \(e(f^* \gamma_n) = f^* e(\gamma_n)\) where \(e(\gamma_n) \in H^n({\operatorname{Gr}}_n({\mathbb{R}}^\infty))\), so \(e\) is the characteristic class defined by \(e(\gamma_n)\).

Let \(E\to X\) be an orientable bundle, then

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Check that \(\alpha \smile\beta = (-1)^{ij} \beta \smile\alpha\).

The Euler class is the obstruction to finding a nonvanishing section over the \(n{\hbox{-}}\)skeleton of \(X\). Note that if we have a line bundle, we automatically have a section. Conversely, a nonvanishing section will produce a line bundle summand.

Some review: let \(X\in {\mathsf{CW}}\) and write \(H^*(X; {\mathbb{Z}})\) for cellular homology. Then \(C^k_{{ \mathrm{cell}} }(X) = \mathop{\mathrm{Hom}}(H_k(X^{(k)}, X^{(k-1)} ), {\mathbb{Z}}) = \mathop{\mathrm{Hom}}(C_k^{{ \mathrm{cell}} }, {\mathbb{Z}})\). To produce the differential (green), use that the LES for the pairs (red and blue) are intertwined:

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Thus we have a formula \begin{align*} {\left\langle { \delta \alpha},~{ [\Delta^{k+1}] } \right\rangle} = {\left\langle { \alpha},~{ {{\partial}}[\Delta^{k+1}] } \right\rangle} .\end{align*}

Recall that if the fibers of a bundle were \(n{\hbox{-}}\)connected, we could construct sections on \(X^{(n-1)}\). Given any section, we can use a Riemannian metric to project onto norm 1 elements to get a section for the sphere bundle, say \(s: X^{(n-1)} \to {\mathbb{S}}E\). Note that \(\pi_i(S^{n-1}) = 0\) for \(i\leq n-2\), so we can construct such a section. Let \(i: \Delta^n\to X\) be the attaching map for some \(n{\hbox{-}}\)cell, then form the following pullback to get a trivial bundle, where we can pull back the section:

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So \(s\) yields a map \({{\partial}}\Delta^n \cong S^{n-1} \to S^{n-1}\), so it has a degree. Then a cocycle can be defined as \([\Delta^n] \xrightarrow{e_s} \deg( {{\partial}}\Delta^n \to S^{n-1})\in {\mathbb{Z}}\).

Recall that the Euler class is the obstruction to finding a nowhere vanishing section on the \(n{\hbox{-}}\)skeleton. Given \(E\to X \in { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }(X)^{\dim = n}\), we can form the sphere bundle \(S^{n-1}\to {\mathbb{S}}E \to X\). Define a section \(s: X\to {\mathbb{S}}E\) over \(X{ {}^{ (0) } }\), then inductively if \(s\) is defined over \(X{ {}^{ (i-1) } }\) for \(i<n\), let \(j: \Delta^i \to X{ {}^{ (i) } }\) be the attaching map for an \(i{\hbox{-}}\)cell. Note that \(\Delta^i\) is contractible, and we can form a pullback

If \(s\) is defined over \(X{ {}^{ (i-1) } } \subseteq X{ {}^{ (i) } }\), we have \({ \left.{{s}} \right|_{{{{\partial}}\Delta^i}} }: {{\partial}}\Delta^i \to S^{n-1}\). But \({{\partial}}\Delta^{i} \cong S^{i-1}\), and if \(i\leq n-1\), \(\pi_i S^{n-1} = 0\) so \(s\) extends over \(\Delta^i\). If \(i=n\), note that \(\pi_{n-1} S^{n-1} \cong {\mathbb{Z}}\), so we define \(e_s(\Delta^n)\) to be the corresponding homotopy class of \(s\), i.e. \(e_s(\Delta^n)\) is the degree of this map between spheres.

We showed that

• \(e_s\) doesn’t depend on the trivialization chosen, and
• If \(s'\) is another nowhere vanishing section on \(X{ {}^{ (n-1) } }\), then \(e_s - e_{s'} \in \operatorname{im}\delta\) for \(\delta: C^{n-1}(X) \to C^n(X)\).
• If \(\delta e_s = 0\), then \([e_s] \in H^n(X; {\mathbb{Z}})\). The content here is that the cochain descends to a cocycle.

For \(i:\Delta^{-1} \to X{ {}^{ (n-1) } }\) and \({\mathbb{S}}E\to X{ {}^{ (n-1) } }\), form the pullback \(i^* {\mathbb{S}}E\). Generally given two sections \(s, s': \Delta^{n-1} \to i^* {\mathbb{S}}E \cong \Delta^{n-1} \times S^{n-1}\), we can glue them to define a map \(-\Delta^{n-1} {\textstyle\coprod}_{{{\partial}}} \Delta^{n-1} \xrightarrow{-s {\textstyle\coprod}s'} S^{n-1}\). The glued space is an \(S^{n-1}\), so the degree of this map will be \(\alpha^{\Delta^{n-1}}\)

Recall that a smooth structure on a manifold \(M\) is a collection \((U, \phi_U)\) where \(U \subseteq M\) is open, \(\phi_U: U\to {\mathbb{R}}^n\) is a homeomorphism onto an open subset of \({\mathbb{R}}^n\), and for all \(U, V\) we have \begin{align*} \psi_{VU} \mathrel{\vcenter{:}}=\phi_V^{-1}\circ \phi_U: \phi_U(U \cap V) \to \phi_V(U \cap V) \in C^\infty ,\end{align*} so there are all diffeomorphisms of open subsets of \({\mathbb{R}}^n\). A smooth atlas is maximal if not contained in any other smooth atlas, and two atlases are compatible if their union is again a smooth atlas. We say two smooth structures are equivalent if they are compatible.

Recall that for \(f: {\mathbb{R}}^n\to {\mathbb{R}}^m\), \(df\in \operatorname{Mat}(m\times n; {\mathbb{R}})\) is given by \((df)_{ij} = {\frac{\partial f_i}{\partial x_j}\,}\). For any \(p\in U \cap V\), we have \(d_{\phi_U(p)} \psi_{VU} \in \operatorname{GL}_n({\mathbb{R}})\), so we get a map \(d\psi_{VU}: U \cap V\to \operatorname{GL}_n({\mathbb{R}})\). By the chain rule they satisfying the cocycle definition, so these glue to a vector bundle \({\mathbf{T}}M\to M\).

Some background from smooth manifolds: a map \(f:M\to N\) of manifolds is smooth if for any smooth charts \((U, \phi_U)\) and \((V, \psi_V)\) on \(M, N\) respectively, the transition map \(\psi_V \circ f \circ \phi_U ^{-1}{\mathbb{R}}^m\to{\mathbb{R}}^n\) is smooth.

\(N \subseteq M\) is a smooth submanifold if for any \(p\in N\) there exists a smooth chat \((U, \varphi)\) on \(M\) such that \(p\in U\) and \(\phi(U \cap N) \subseteq {\mathbb{R}}^k \times\left\{{0}\right\} \subseteq {\mathbb{R}}^n\).

A Lie group is a group \(G\) with the structure of a smooth manifold where multiplication and inversion are smooth self-diffeomorphisms.

Given a connection, curves \(\gamma\) in \(M\) can be lifted to curves \(\tilde \gamma\) in \(P\) in such a way that tangents of \(\tilde \gamma\) are projected to tangents of \(\gamma\):

Consider \(\mathop{\mathrm{Frame}}({\mathbf{T}}M)\), then recall that \({{\mathbf{T}}M} = \mathop{\mathrm{Frame}}({\mathbf{T}}M) \overset{\scriptscriptstyle {\operatorname{GL}_n({\mathbb{R}})} }{\times} {\mathbb{R}}^n\) by the mixing construction. Given a connection on \(\mathop{\mathrm{Frame}}({\mathbf{T}}M)\), we can parallel transport vectors in \({\mathbf{T}}M\) along curves. This comes from taking \([F_0, v_0]\) and evolving \(F_0\) along \(F_t \mathrel{\vcenter{:}}=\tilde \gamma(t)\), choosing pairs \([F_t, v_0]\) for all \(t\), and ending at \([F_t, v_0]\).

Today: computations involving the Euler class.

Given \(N^n \hookrightarrow M^m\) an oriented closed submanifold with \(M\) closed, we can consider Thom class of the disc/sphere bundles. We identify the disc bundle \({\mathbb{D}}\nu N\) with a tubular neighborhood of \(N\) in \(M\) and apply excision to get the following:

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We can also consider the composition:

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An aside on cup/cap products. The cap product is a map: \begin{align*} \frown: H_m(X) \otimes_{\mathbb{Z}}H^i(X) &\to H_{m-i}(X) \\ .\end{align*} On chains, it’s given by \begin{align*} \frown C_m(X) \otimes_{\mathbb{Z}}C^i(X) &\to C_{m-i}(X) \\ (\sigma, \phi) &\mapsto \phi\qty{ { \left.{{\sigma}} \right|_{{{\left[ {v_0,\cdots, v_i} \right]}}} } } { \left.{{\sigma}} \right|_{{v_{i+1}, \cdots, v_m}} } .\end{align*} Then \(\mathrm{PD}({-}) \mathrel{\vcenter{:}}=[X]\frown({-})\).

The cup product is a map \begin{align*} \smile: H^i(X) \otimes_{\mathbb{Z}}H^j(X) &\to H^{i+j}(X) \\ C^i(X) \otimes_{\mathbb{Z}}C^j(X) &\to C^{i+j}(X) \\ (\phi, \psi) &\mapsto \qty{\sigma \mapsto \phi\qty{{ \left.{{\sigma}} \right|_{{[v_0,\cdots, v_i]}} }} \psi\qty{{ \left.{{\sigma}} \right|_{{[v_{i+1}, \cdots, v_{i+j}}} }} } .\end{align*} Then fixing any element yields a map \(\alpha \smile({-}): C^j(X) \to C^{i+j}(X)\), which is induced by a map \(\phi\frown({-}): C_{i+j}(X) \to C_i(X)\).

Recall that if \(N^n \leq M^n\) is a submanifold, we have the following diagram:

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Then the Thom class \(u_{\nu N}\in H^{m-n}({\mathbb{D}}\nu N, {\mathbb{S}}\nu N) \to H^{m-n}({\mathbb{D}}\nu N_x, {\mathbb{S}}\nu N_x)\) is mapped to the generator specified by the orientations on fibers.

Then \begin{align*} [( {\mathbb{D}}\nu N_x, {\mathbb{S}}\nu N_x) \frown N] = [\left\{{x}\right\}] ,\end{align*} which is the positive generator. So \begin{align*} \mathrm{PD}[ ({\mathbb{D}}\nu N_x, {\mathbb{S}}\nu N_x) ] \smile\mathrm{PD}[N] = \mathrm{PD}[x] .\end{align*} Now we can cap this to obtain \begin{align*} [{\mathbb{D}}\nu N, {\mathbb{S}}\nu N] \frown\qty{ \mathrm{PD}[ ({\mathbb{D}}\nu N_x, {\mathbb{S}}\nu N_x) ] \smile\mathrm{PD}[N]} &= \qty{ [{\mathbb{D}}\nu N, {\mathbb{S}}\nu N] \frown\mathrm{PD}[{\mathbb{D}}\nu N_x, {\mathbb{S}}\nu N_x] } \frown\mathrm{PD}[N] \\ &= [{\mathbb{D}}\nu N, {\mathbb{S}}\nu N] \frown\mathrm{PD}[x] \\ &= 1 ,\end{align*} where we’ve used that \({\left\langle {\mathrm{PD}[x]},~{ [{\mathbb{D}}\nu N, {\mathbb{S}}\nu N] } \right\rangle}\). So \(\mathrm{PD}[N]\) is \(j^*\) of the Thom class of \(\nu N\).

Recall that \(s^{-1}(0) \leq M\) is a smooth submanifold, and \({ \left.{{ds}} \right|_{{Z}} }: \nu Z { { \, \xrightarrow{\sim}\, }}{ \left.{{E}} \right|_{{Z}} }\), and since this orients \(\nu Z\) this orients \(Z\) as well.

The Euler class for a bundle \(E\to X\in {\mathsf{CW}}\) is the obstruction to finding a nowhere vanishing section on \(X{ {}^{ (n) } }\) for \(n\mathrel{\vcenter{:}}=\dim E\), and \(e(E) = 0\) iff there is a nowhere vanishing section on \(X{ {}^{ (n) } }\). For a smooth manifold \(M\) and \(E \xrightarrow{\pi} M\) with \(\dim M = n, \dim E = k\) and \(s:M\to E\) a section, then \(\mathrm{PD}[s^{-1}(0)] = e[E]\) since \(\dim s^{-1}(0) = n-k\). If \(E = {\mathbf{T}}M\), then \(e({\mathbf{T}}M) = 0\) implies \(\chi(M) = 0\) and there exists a nowhere vanishing vector field.

For \(F\to E \xrightarrow{\pi} B\) a fiber bundle with a \(G{\hbox{-}}\)structure, we have maps:

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A vector bundle was an \({\mathbb{R}}^n\) bundle with a \(\operatorname{GL}_n{\hbox{-}}\)structure, having a Riemannian metric meant having an \({\operatorname{O}}_n\) structure, and orientability meant having a \(\operatorname{GL}^+_n\) structure. A principal \(G{\hbox{-}}\)bundle was \(P\to B\) where \(P\) has a \(G{\hbox{-}}\)action acting freely and transitively on each fiber. Taking the frame bundle sent vector bundles to principal \(\operatorname{GL}_n{\hbox{-}}\)bundles.

We had a construction sending fiber bundles to principal \(G{\hbox{-}}\)bundles, namely the clutching construction: given \({\mathcal{U}}\rightrightarrows X\) and \(\phi_{ij}: U_i \cap U_j \to G\) satisfying the cocycle condition \(\phi_{ij} \phi_{jk} = \phi_{ik}\), we get a fiber bundle with fiber \(F\) and transition functions for any \(F\in{G{\hbox{-}}\mathsf{Spaces}}\).

On universal bundles and classifying spaces: for \(X\in {\mathsf{CW}}\), \begin{align*} \mathop{\mathrm{Prin}}{ {\mathsf{Bun}}\qty{G} }(X) { { \, \xrightarrow{\sim}\, }}[X, {\mathbf{B}}G] \\ f^* EG &\mapsfrom f .\end{align*}

We noted

• \(G\) discrete implies \({\mathbf{B}}G \simeq K(G, 1)\)
• \(K(C_2, 1) = {\mathbb{RP}}^{\infty}\) with \(EC_2 = S^{\infty}\)
• \({\mathbf{B}}U_1 = {\mathbb{CP}}^{\infty}\) with \(EU_1 = S^{\infty}\)
• \({{\mathbf{B}}{\operatorname{O}}}_n = {\operatorname{Gr}}_n({\mathbb{R}}^{\infty})\), with \({\mathsf{E} {\operatorname{O}}}_n = V_n({\mathbb{R}}^{\infty})\) the Stiefel manifold of \(n{\hbox{-}}\)dimensional frames.
• \({\mathbf{B}}{\operatorname{SO}}_n\) are oriented \(n{\hbox{-}}\)planes in \({\mathbb{R}}^\infty\).
• For \(H\leq G\), \(EH = EG\) and \({\mathbf{B}}H = EG/H\).

We had a canonical bundle \(\gamma_n \to {\operatorname{Gr}}_n({\mathbb{R}}^\infty)\) whose fiber above \(W\leq {\mathbb{R}}^{\infty}\) was exactly \(W\), so \(\gamma_n = \left\{{(W, w\in W) \subseteq {\operatorname{Gr}}_n({\mathbb{R}}^\infty) \times {\mathbb{R}}^{\infty}}\right\}\). Every vector bundle \(E\to X\) is of the form \(f^* \gamma_n\to X\) for some \(f\in [X, {\operatorname{Gr}}_n({\mathbb{R}}^\infty)]\), and similarly \({\operatorname{O}}_n\) bundles are pullbacks of \(V_n({\mathbb{R}}^\infty)\to {\operatorname{Gr}}_n({\mathbb{R}}^\infty)\).

Note: why \(c_1=0\) in symplectic settings, related to Maslov index and ensures that the dimension of the relevant moduli space is zero.

The Euler class is natural in the following sense: for \(E\to X\) with \(\dim E = n\) and \(X\in {\mathsf{CW}}\), we can write

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Then naturality is the following equality: \begin{align*} e(E) = e(f^* \gamma_n) = f^*(e(\gamma_n)) .\end{align*}

Recall the Thom isomorphism theorem: for an oriented \(E\in { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }^n\), we have a disk bundle \({\mathbb{D}}E\) and sphere bundle \({\mathbb{S}}E\), and \begin{align*} H^j({\mathbb{D}}E; {\mathbb{S}}E) \cong \begin{cases} 0 & j< n \\ {\mathbb{Z}}& j=n \\ H^{j-n}({\mathbb{D}}E) & j> n \end{cases} ,\end{align*} noting that we can take \(H^n({\mathbb{D}}E, {\mathbb{S}}E) \to H^n ({\mathbb{D}}E_x, {\mathbb{S}}E_x) { { \, \xrightarrow{\sim}\, }}H^n(D^n, S^{n-1})\), where the target has a canonical positive generator. The preimage of this generator is \(u_E\), the Thom class. The isomorphisms in the range \(j>n\) are given by \({-}\smile u_E\).

We had a claim: \begin{align*} H^n({\mathbb{D}}E, {\mathbb{S}}E) &\to H^n({\mathbb{D}}E) \cong H^n(X) \\ u_E &\mapsto e(E) .\end{align*}

On the Gysin sequence: use the bundle \(S^{n-1} \hookrightarrow{\mathbb{S}}E \to X\) and the LES \begin{align*} \cdots \to H^{j-1}({\mathbb{S}}E) \xrightarrow{\delta} H^{j-n}(X) \xrightarrow{({-}) \smile e(E)} H^j(X) \to H^j({\mathbb{S}}E) \to \cdots .\end{align*} The connecting map \(\delta\) comes from the Thom isomorphisms \(H^j({\mathbb{D}}E, {\mathbb{S}}E){ { \, \xrightarrow{\sim}\, }}H^{j-n}({\mathbb{D}}E)\) and splicing the LES of the pair \(({\mathbb{D}}E, {\mathbb{S}}E)\).

So \(e(E)\) is the obstruction to finding a nonvanishing section over the \(n{\hbox{-}}\)skeleton, where \(\dim E = n\). Given a nonvanishing section over \(X{ {}^{ (k-1) } }\), consider extending it over \(X{ {}^{ (k) } }\). We can use the cellular attaching maps to write

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• If \(s \in \pi_k(S^n)\) and \(k<n\), it is nullhomotopic and can thus be extended from \({{\partial}}\Delta^k\) to \(\Delta^k\).
• If \(k=n\), we get \(s_*: H_{n-1}({{\partial}}\Delta^n) \to H_{n-1}(S^{n-1})\) where \(a\mapsto na\) with \(n\mathrel{\vcenter{:}}=\deg(s_*)\). This yields an cellular \(k{\hbox{-}}\)cochain \(C^k \to {\mathbb{Z}}\) where \(\Delta^n \mapsto \deg s_*\), and this represents the Euler class.

For smooth manifolds, we have natural bundles \({\mathbf{T}}X\) and \(\nu X\) to work with. Then \begin{align*} {\left\langle {e({\mathbf{T}}M)},~{ [M] } \right\rangle} = \chi(M) ,\end{align*} using that \(e({\mathbf{T}}M) \in H^n(M) \cong {\mathbb{Z}}\). In general, so \(E\to M^n\) with a generic section \(s\), writing \(Z\mathrel{\vcenter{:}}= s^{-1}(0) \leq E\) a submanifold of dimension \(n-k\) when \(\dim Z = k\), we have \([Z] \in H_{n-k}(M)\) and \(\mathrm{PD}[Z]= e(E)\).

For \(N^n \subseteq M^m\), we have

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So we can compute \(e(E)\) for \({\mathbf{T}}M\) and \(\nu N\).

Next topics: Chern and Stiefel-Whitney classes.

Next time: existence and uniqueness.

The Stiefel-Whitney classes \(w_i \in H^i(X; C_2)\) will be defined for \({ {\mathsf{Bun}}\qty{\operatorname{GL}_r} }(X)_{/ {{\mathbb{R}}}}\) while Chern classes \(c_i \in H^{2i}(X; {\mathbb{Z}})\) will be defined for \({ {\mathsf{Bun}}\qty{\operatorname{GL}_r} }(X)_{/ {{\mathbb{C}}}}\). We setting \(w(E) \mathrel{\vcenter{:}}=\sum_i w_i(E) \in H^*(X; C_2)\), we mentioned several properties:

• \(w_i(f^* E) = f^* w_i(E)\)
• \(w(E_1 \oplus E_2) = w(E_1) \smile w(E_2)\)
• \(w_{> \dim E} E = 0\)
• \(w_1(\gamma)\) is the generator of \(H^1({\mathbb{RP}}^\infty; C_2) = C_2\) where \(\gamma \to {\mathbb{RP}}^\infty\) is the canonical line bundle. For complex bundles, \(c_1(\gamma)\) is the positive generator of \(H^2({\mathbb{CP}}^\infty; {\mathbb{Z}}) = {\mathbb{Z}}\).

These properties characterize the \(w_i\) and \(c_i\) uniquely.

On why we need \(C_2\) coefficients: we’re pulling back \(\gamma \to {{\mathbf{B}}{\operatorname{O}}}_1\) and \({{\mathbf{B}}{\operatorname{O}}}_1 \simeq{\mathbb{RP}}^\infty\) where \(H^1({\mathbb{RP}}^\infty; {\mathbb{Z}}) = 0\)! For line bundles, we’ll automatically have \(w_{\geq 2} = 0\) for any such class, so we only have \(w_1\) the work with, and this we need to pull back something nonzero to get anything interesting.

Consider inverting a formal power series \(\sum_{i\geq 0} a_i\): \begin{align*} (1 + a_1 + a_2 + a_3 + \cdots)( 1 - a_1 + (a_1^2 + a_2) + (a_1^3 + 2a_1 a_2 - a_3) + \cdots) = 1 .\end{align*}

So if \(w(E_1)w(E_2) = 1\), we can solve for \(w(E_1)\) in terms of \(w(E_2)\).

Recall that \(w({\mathbf{T}}{\mathbb{RP}}^n) = (1+a)^{n+1}\) where \(\left\langle{a}\right\rangle = H^1({\mathbb{RP}}^n; C_2)\), and as an application, if \({\mathbb{RP}}^{2^r} \hookrightarrow{\mathbb{R}}^N\), then \(N \geq 2^{r+1} - 1\). Similarly, \(c({\mathbf{T}}{\mathbb{CP}}^n) = (1+a)^{n+1}\) with \(\left\langle{a}\right\rangle = H^2({\mathbb{CP}}^\infty; {\mathbb{Z}})\) the positive generator.

Write \(c(E) = 1 + c_1(E) + \cdots + c_n(E)\), so \begin{align*} \pi^* c(E) &= c(\gamma) c(E') = (1 +c_1(\gamma))(1 + c_1(E') + \cdots + c_{n-1}(E'))\\ &= 1 + (c_1(\gamma) + c_1(E')) + (c_1( \gamma)c_1(E') + c_2(E')) + \cdots + (c_1(\gamma) c_{i-1}(E') + c_1(E') ) .\end{align*}

Plugging this into the LHS above yields \begin{align*} c_1( \gamma {}^{ \vee })^n + (c_1(\gamma) + c_1(E')) c_1(\gamma {}^{ \vee })^{n-1} + (c_1(\gamma)c_1(E') + c_2(E'))c_1(\gamma {}^{ \vee })^{n-2} + \cdots + c_1(\gamma)c_{n-1}(E') &= (c_1 (\gamma) + c_1(\gamma {}^{ \vee }))( c_1( \gamma {}^{ \vee })^{n-1} + c_1(E') c_1( \gamma {}^{ \vee })^{n-2} + c_2(E') c_1( \gamma {}^{ \vee })^{n-2} + \cdots + c_{n-1}(E') ) .\end{align*} Since \(\gamma {}^{ \vee }\) and \(\gamma\) are dual, the first term is zero, so this entire expression is zero.

Consider the following pullbacks:

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What is \(f^* E\)? There is a pullback

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Here \(\phi^* \gamma\) will be trivial: we have \(\gamma \subseteq {\mathbb{CP}}^{n-1} \times {\mathbb{C}}^n\), so \(\phi^* \gamma \subseteq S^{2n-1} \times {\mathbb{C}}^n\) and \(\dim_{\mathbb{R}}\phi^* \gamma = 2\). Then the fibers are \(\phi^*\gamma_x = \left\{{(x, v) {~\mathrel{\Big\vert}~}v = \lambda x,\, \lambda \in {\mathbb{C}}}\right\}\). It turns out that \(\phi^* E = {\mathbb{C}}\oplus q^* E'\) where \({\mathbb{S}}E \xrightarrow{q} {\mathbb{P}}(E)\). Writing \(\pi: {\mathbb{P}}(E) \to X\), we have \begin{align*} (\pi \circ q)^* c_i(E) = c_i(E) && i < n-1 ,\end{align*} and \(c_n(E) = e(E)\). To find this we’ll need \(\pi \circ q\) to be injective.

Consider

By the Gysin sequence, for \(j<2n-1\) we’ll have \(H^j(X) \cong H^j({\mathbb{S}}E)\) since the red terms vanish:

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This uses that for \(j<2n\) that \(H^{j-2n}(X) = 0\). For \(i\leq n-1\), \(2i<2n-i<2n-1\), and so the Chern classes for \(X\) and \({\mathbb{S}}E\) are isomorphic via \((\pi \circ q)^*: H^j(X) \to H^j({\mathbb{S}}E)\) in the LES. By induction, we can define \(c_i(E)\) for \(i\leq n-1\).

Idea: pull back to split off a line bundle, pull back further to split off another line bundle, and continue. This is why we only need \(c_1\) to determine a line bundle.

Today: proving/checking the axioms for Chern classes. Recall that given \(E \xrightarrow{\pi} X\), we can form a pullback:

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Here \(\gamma {}^{ \vee }\) is the dual of \(\gamma\) and \(c_1( \gamma {}^{ \vee }) = -_1( \gamma)\). By Leray-Hirsch, we have \begin{align*} 1 + c_1( \gamma {}^{ \vee }) + c_1(\gamma {}^{ \vee })^2 + \cdots + c_1( \gamma {}^{ \vee })^{n-1} \divides (c_1(\gamma {}^{ \vee }))^n \end{align*} and for \(c_i(E)\) the \(i\)th Chern class of \(E\), \begin{align*} c_1(\gamma {}^{ \vee })^n + c_1(E) c_1(\gamma {}^{ \vee })^{n-1} + \cdots + c_n(E) = 0 .\end{align*}

Recall the alternative description of Chern classes:

Link to Diagram

By the Gysin sequence, we obtained \(H^j(X) \cong H^j({\mathbb{S}}E)\) for \(j\leq n-2\), and there are \(c_1(E'), \cdots, c_{n-2}(E')\) living in \(H^2({\mathbb{S}}E), H^4({\mathbb{S}}E), \cdots, H^{2n-2}({\mathbb{S}}E)\). So we defined \(c_i(E)\) such that \(\pi^* c_i(E) = c_i(E')\) for \(i\leq n-1\), so \(c_n(E) = e(E)\).

Recall that we were proving the splitting principle: given \(E\to X\) of dimension \(n\), there is a space \(Y\) with \(f:Y\to X\) where

• \(f^* E = \bigoplus L_i\) splits as a direct sum of line bundles.
• \(f^*: H^*(X) \to H^*(Y)\) is injective.

Recall that the Euler class is the obstruction to finding an obstruction to extending a section over the \(n{\hbox{-}}\)skeleton.

Why? For example, consider \(E\to S^1\). Trivialize over \(S^1\setminus\left\{{{\operatorname{pt}}}\right\}\), then glue the ends by some element \(A\in \operatorname{GL}_n({\mathbb{R}})\). If \(\operatorname{det}(A) > 0\), this will be orientable, and \(\operatorname{det}(A) < 0\) will be nonorientable.

For \(k=n\), \(w_n(E)\) corresponds to 1 linearly independent section of \(X{ {}^{ (n) } }\) since \(e(E) \equiv w_n(E) \operatorname{mod}2\). For \(k=1\), \(w_1(E)\) corresponds to \(n\) linearly independent sections over \(X{ {}^{ (1) } }\).

In terms of transition functions (i.e. Čech cohomology data), this is the inclusion \(\operatorname{GL}_n({\mathbb{R}}) \hookrightarrow\operatorname{GL}_n({\mathbb{C}})\). We can now consider the Chern classes \(c_i(E\otimes{\mathbb{C}})\).

Note that \(E\otimes{\mathbb{C}}\cong E \oplus E\), and we can map \begin{align*} {\mathbb{R}}{ {}^{ \scriptscriptstyle\oplus^{2} } }\otimes{\mathbb{C}}&{ { \, \xrightarrow{\sim}\, }}{\mathbb{C}}{ {}^{ \scriptscriptstyle\oplus^{2} } } \\ (x\oplus y) \otimes\left\langle{1, i}\right\rangle &\mapsto (x, ix) \oplus (y, iy) .\end{align*}

Suppose \(R\) is a coefficient ring where 2 is invertible (or not a zero divisor). Some properties of the \(p_i\):

1. \(p_i\) is natural, so \(p_i (f^* E) = f^* p_i(E)\).

2. \(p_{i> \dim(E)/2}\).

3. \(p(E \bigoplus E') = p(E) \smile p(E')\). Why? Note \(p(E) = 1 - c_2 + c_4 - c_6 + \cdots\), so multiplying two such things yields \begin{align*} p(E) p(E') = 1 - (c_2 + c_2') + (c_4 + c_2c_2' + c_4') - (c_6 + c_4 c_2' + c_2 c_4' + c_6') - \cdots = p(E \oplus E') .\end{align*}

4. If \(\dim_{\mathbb{R}}E = 2n\) then \(p_n(E) = e(E)^2\). Why? \begin{align*} p_n(E) = (-1)^n c_{2n}(E\otimes{\mathbb{C}}) = (-1)^n e(E\otimes{\mathbb{C}}) = (-1)^n (-1)^{2n(2n+1) \over 2} = (-1)^{n + n(2n+1)} e(E)^2 = (-1)^{n(2n+2)}e(E)^2 = e(E)^2 .\end{align*}

5. If \(E\) is a complex line bundle, \begin{align*} c(E \oplus \mkern 1.5mu\overline{\mkern-1.5muE\mkern-1.5mu}\mkern 1.5mu) = c(E\otimes{\mathbb{C}}) = 1 - p_1(E) + p_2(E) - \cdots = c(E) c(\mkern 1.5mu\overline{\mkern-1.5muE\mkern-1.5mu}\mkern 1.5mu) = (1 + c_1(E) + c_2(E) + \cdots)(1 - c_1(E) + c_2(E) - \cdots) .\end{align*}

This defines an equivalence relation on \({\mathsf{sm}}{\mathsf{Mfd}}^n\), and yields a ring \((\prod_n {\mathcal{N}}^n, {\textstyle\coprod}, \times)\). We’ll add extra structure to get refinements of this ring, for which we’ll need a bit about stable normal bundles and the Whitney embedding theorem.

Moreover if \(i, i': M\hookrightarrow{\mathbb{R}}^n\) with \(n\geq 2n+3\), then the normal bundles \(\nu M, (\nu M)'\) are bundle isomorphic.

Note that adding trivial line bundles yields a tower \({{\mathbf{B}}{\operatorname{O}}}_1 \to {{\mathbf{B}}{\operatorname{O}}}_{2}\to \cdots\), where the classifying maps of \(E\) and \(E \oplus L\) are classified by maps to \({{\mathbf{B}}{\operatorname{O}}}_k\) and \({{\mathbf{B}}{\operatorname{O}}}_{k+1}\) respectively. We can define lifts of extra structures by looking for commutative towers over \(X_k\) for e.g. \(X = {{\mathbf{B}}{\operatorname{O}}}, {\operatorname{Spin}}\), etc:

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For orientations, we get a bordism ring \(\Omega_n\), and for framings we get \(\Omega_n^{\mathop{\mathrm{Fr}}}\).

For \(n=3\), the only possibilities for \(I\) are

• \((3,0,0) \leadsto W_n = w_1^3({\mathbf{T}}M)\)
• \((1, 1, 0 \leadsto W_n = w_1({\mathbf{T}}M)w_2({\mathbf{T}}M) )\)
• \((0,0,1) \leadsto W_n = w_3({\mathbf{T}}M)\)

Milnor and Wall: the only torsion in \(\Omega_n\) is order 2, and an oriented manifold \(M\) is 0 in \(\Omega_n \iff w_I(M), P_I(M) = 0\) for all \(I\). For a proof (for at least the first statement), see Milnor-Stasheff.

We can kill torsion to get an injective map \begin{align*} \Omega_n \otimes{\mathbb{Q}}\xrightarrow{(P_{I_1}, \cdots, P_{i_k} ) } {\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{k} } } .\end{align*}

Today: the Hirzebruch signature theorem and exotic \(S^7\). We saw that \(\Omega_* \otimes{\mathbb{Q}}\) is the polynomial algebra on \({\mathbb{Q}}[v_1, v_2\cdots]\) where \(v_2\) corresponds to \({\mathbb{CP}}^{2n}\). This was because \(\ker\qty{\Omega_n \xrightarrow{\left\{{p_{I_n}}\right\}} {\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{n} } } }\) can only be torsion.

Let \(C \mathrel{\vcenter{:}}= B_1 { \displaystyle\coprod_{M} } B_2\) where \({{\partial}}B_i = M\), then there is a relation expressing \({\left\langle {p_1(C)^2},~{[C]} \right\rangle}\) in terms of pairings of \(p(B_i)\) against \([B_i]\), and \(\sigma(C) = \sigma(B_1) - \sigma(B_2)\). We have \begin{align*} 45\sigma(C) + {\left\langle {p_1(C)^2},~{[C]} \right\rangle} &\equiv 0 \operatorname{mod}7 \\ \implies -40\sigma(C) + {\left\langle {p_1(C)^2},~{[C]} \right\rangle} &\equiv 0 \operatorname{mod}7 \\ \implies \sigma(C) + 2{\left\langle {p_1(C)^2},~{[C]} \right\rangle} &\equiv 0 \operatorname{mod}7 \\ \implies (\sigma(B_1) - \sigma(B_2) ) - 2 \qty{ {\left\langle {(i^{-1}p_1 B_1)^2 },~{[B_1]} \right\rangle} \cdots } &= \lambda_{B_1}(M) - \lambda_{B_2}(M) .\end{align*}

If the \(\lambda\)s differ, the manifolds can not be diffeomorphic. Constructing \(S^7\)s: take \(S^3\) bundles over \(S^4\). Pick any map \(S^3\hookrightarrow{\operatorname{SO}}_4({\mathbb{R}}) \subseteq {\operatorname{Homeo}}(S^3)\) to get clutching data, which are elements of \(\pi_3({\operatorname{SO}}_4) = {\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{2} } }\). Label these bundles with parameters \((m, n) \in {\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{2} } }\) so that \(\xi_{1, 0}\) corresponds to lifting \([S^3, {\operatorname{SO}}_3]\) to \([S^3, {\operatorname{SO}}_4]\) fixing the \(x{\hbox{-}}\)axis, and \(\xi_{1, 1}\) is the canonical for \({\mathbb{H}}{\mathbb{P}}^1\). If \(M_k\) is the total space of \(\xi_{k, 1}\), then \(\lambda(M_k) \equiv k^2-1 \operatorname{mod}7\), so aren’t diffeomorphic. Then one can show that the \(M_k\) are homeomorphic by finding a Morse function with exactly 2 critical points.

Definition of vector bundle: need charts \((U, \phi)\) with \(\phi: \pi^{-1}(U) \to U\times{\mathbb{R}}^n\) which when restricted to a fiber \(F_b\) yields an isomorphism \(F_b { { \, \xrightarrow{\sim}\, }}{\mathbb{R}}^n\).

What are these maps??