Bundle structures and lifting problems (Ex)

From Manifold Atlas

Jump to: navigation, search

[edit] 1 Lifting maps

Given a (pointed) map $\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}g: Y\to Z$ of pointed topological spaces, we define the homotopy fibre of $g$ as

$\displaystyle \mathrm{hofib}(g):=\{(y,\gamma)\in Y\times PZ|\,\gamma(1)=g(y),\,\gamma(0)=z_0\},$ $\displaystyle \mathrm{hofib}(g):=\{(y,\gamma)\in Y\times PZ|\,\gamma(1)=g(y),\,\gamma(0)=z_0\},$

where $PZ$ is the space of paths starting at the base-point of $Z$ . We denote by $p$ : $\mathrm{hofib}(g)\to Y$ the projection.

Exercise 2.1. Prove the following:

A map $f:X \to Y$ of pointed space has a lift $\bar f:X\to \text{hofib}(g)$ along $p$ if and only if $g\circ f$ is homotopic to the constant map.

Hint 2.2. This is a special case of [Hatcher2002, Proposition 4.72].

[edit] 2 Classification of orientations and spin structures on vector bundles

Recall the Definition of Eilenberg-MacLane-spaces. We denote the space of pointed loops in a space $X$ by $\Omega X$ .

Exercise 4.1.

Show: There is a homotopy equivalence $\Omega K(n+1,\Zz/2\Zz)\simeq K(n,\Zz/2\Zz)$ .

Hint: Use the uniqueness of Eilenberg-MacLane-spaces and the long exact sequence in homotopy associated to the path-space-fibration

$\displaystyle \Omega K(n+1,\Zz/2 \Zz)\to P(K(n+1,\Zz/2 \Zz))\to K(n+1,\Zz/2 \Zz) \;.$ $\displaystyle \Omega K(n+1,\Zz/2 \Zz)\to P(K(n+1,\Zz/2 \Zz))\to K(n+1,\Zz/2 \Zz) \;.$

Recall that the path-space is contractible.

Show that the set of homotopy classes of pointed maps $[Y,\Omega X]$ has a group structure induced by composition of paths.

Hint: This is similar to the group structure of the fundamental group.

Maybe you've heard that the group $H^1(X,\Zz/2\Zz)$ acts free and transitively on the set of spin structures of an oriented vector bundle $\xi\to X$ (X a compact pointed space). Now recall that \ref{The group structure on $H^1(X,\Zz/2\Zz)$ is due to the Exercise 3.2.} $H^1(X,\Zz/2\Zz)\cong [X,K(1,\Zz/2\Zz)]$ , where $K(1,\Zz/2\Zz)$ denotes an Eilenberg-MacLane-space.

So we first prove the statement about classification of spin structures. The warm-up is the classification of orientations:

The first Stiefel-Whitney class is a map $w_1:BO\to K(1,\Zz/2\Zz)$ .
The homotopy fiber hofib( $w_1$ ) is $BSO$ .
The projection $p:BSO\to BO$ is the map induced by $SO\hookrightarrow O$ .

Assume that the homotopy groups of $O$ are known.

Calculate the homotopy groups of $BO$ using the fibration $O\to EO\to BO$ .
Calculate the homotopy groups of $BSO$ using the fibration
$\displaystyle BSO\to BO\stackrel{w_1}{\to} K(1,\Zz/2\Zz)\;.$
Calculate the homotopy fibre of $p:BSO\to BO$ .

Now we can classify the orientations on a vector bundle $\xi: E\to X$ . For this we need to know that the sequence

$\displaystyle \text{hofib}(p)\to BSO \to BO$ $\displaystyle \text{hofib}(p)\to BSO \to BO$

fits into the following diagram\footnote{It's non-trivial to see that the functor $B$ can be applied to each of those spaces.}:

$\displaystyle \xymatrix{ \mathrm{hofib}(p) \ar[r]^q \ar[d]^{\simeq} & BSO \ar[r]^p \ar[d]^{\simeq} & BO \ar[d]^{\simeq} \\ \Omega B\mathrm{hofib}(p) \ar[r]^{\Omega Bq} & \Omega BBSO \ar[r]^{\Omega Bp} & \Omega BBO }$

Hence there are group structures on $[X,\text{hofib}(p)]$ and $[X,BSO]$ (the latter one is the Whitney-sum of vector bundles). Furthermore there is an action of $[X,\text{hofib}(p)]$ on $[X,BSO]$ induced by $q$ .

A vector bundle is called orientable if its classifying map $f: X \to BO$ lifts along $p:BSO \to BO$ . An orientation is the choice of such a lift. The first Stiefel-Whitney class of $\xi$ is defined as the composition $w_1\circ f$ , where $w_1$ is defined as above.

Let $X$ denote a compact pointed space and $\xi: E\to X$ a vector bundle on $X$ .

Use Exercise 3.1 to show that $\xi$ is orientable if and only if its first Stiefel-Whitney class vanishes.
Show that the group $[X,\text{hofib}(p)]$ acts free and transitively on the set of homotopy classes of lifts.

Hint: Use the homotopy-lifting property and Exercise 3.1

Give an interpretation of the group $[X,\text{hofib}(p)]$ .

Now there are similar results for spin structures (on oriented vector bundles).

Repeat Exercises 3.3 and 3.4 using the second Stiefel-Whitney class $w_2:BSO\to K(2,\Zz/2\Zz)$ . The homotopy fibre of $w_2$ is BSpin, where Spin is the colimit over $Spin(n)$ , the universal cover of $SO(n)$ .