Main bundle

In mathematics, a G-principal bundle is a special kind of bundle for which the fibers are all principal homogeneous spaces with respect to a topological group.

Major G-bundles are G-bundles in the sense that the G group also serves as the structural group of the bundle. Principal bundles have important uses in topology and differential geometry. They have also found use in physics where they form part of gauge theory. The principal bundles provide a unifying framework in the theory of bundles in the sense that all bundles with structural group G determine a single G-main bundle from which it can be reconstructed the original bundle.

Formal definition

A ${displaystyle G}$-Main fiber is a fiber ${displaystyle pi:Pto X}$ next to a continuous right action ${displaystyle Ptimes Gto P}$ by a topological group ${displaystyle G}$ such as ${displaystyle G}$ preserves fibers ${displaystyle P}$ and action is free and transitive. Abstract fiber fiber is taken as ${displaystyle G}$ (often it is required that the base space ${displaystyle X}$ be a Hausdorff space and possibly paracompact).

The orbits of the ${displaystyle G}$-action is precisely the fibers of the fiber and the orbit space is homeomorph to the homogeneous space ${displaystyle P/G}$.

A ${displaystyle G}$-Main fibre can also be characterized as a ${displaystyle G}$-fiber. ${displaystyle pi:Pto X}$ fiber ${displaystyle G}$ where the group of the structure acts in the fiber by multiplication to the left. Since multiplication on the right by ${displaystyle G}$ in the conmute fiber with the action of the structural group, there is an invariant notion of multiplication to Right ${displaystyle G}$ on ${displaystyle P}$.

The notion of primary fiber can be extended to the category of differential variations, requiring that ${displaystyle pi:Pto X}$ be a differential application between varieties, ${displaystyle G}$ a group of Lie and that the action of ${displaystyle G}$ on ${displaystyle P}$ be different.

Examples

The most common example of a differential main fiber is the reference fiber, also called frame fibre, of a variety ${displaystyle M}$. The fiber on a point ${displaystyle xin M}$ is the system of all references (i.e. ordered bases) of the space tangent ${displaystyle T_{x}M}$. The general linear group ${displaystyle GL(n,mathbb {R}}}}$ acts in a simple and transient way on the set of bases. These fibers can be unified naturally to obtain a ${displaystyle GL(n,mathbb {R}}}}$-Main work on ${displaystyle M}$.

Variations in the previous example include the fiber of orthonormal references of a riemannian variety. Here the references should be orthonormal bases regarding the metric. The structural group is the orthogonal group ${displaystyle O(n)}$.

Yeah. ${displaystyle X}$ It's a topological space. ${displaystyle p:Cto X}$ is a normal (regular) covering, the latter can be considered a main fiber where the structural group ${displaystyle pi _{1}(X)/p_{*}pi _{1}(C)}}$ acts on ${displaystyle C}$ through the action of monodromy. In particular, the universal covering of a topological space ${displaystyle X}$ is a main fiber over ${displaystyle X}$ structural group ${displaystyle pi _{1}(X)}$.

Sean. ${displaystyle G}$ a group of Lie and ${displaystyle H}$ a closed subgroup (not necessarily normal). Then ${displaystyle G}$ It's a ${displaystyle H}$-Main fiber over the quotient space (left) ${displaystyle G/H}$. Here. action ${displaystyle H}$ in ${displaystyle G}$ is multiplication to the right. Fibers are the left-hand conjuncts of ${displaystyle H}$ (in this case there is a distinguished fiber, which contains the identity, which is naturally isomorph to ${displaystyle H}$).

Consider the projection ${displaystyle pi:S^{1}to S^{1}}$given ${displaystyle zmapsto z^{2}}$. East ${displaystyle mathbb {Z} _{2}}$-Main fiber is associated fiber to the Moebius band.

The projective spaces provide interesting examples of major fibers. Let's remember that ${displaystyle n}$-Sphere ${displaystyle S^{n}}$is a double covering of real projective space ${displaystyle mathbb {R} mathbb {P} ^{n}}$. The natural action of ${displaystyle O(1)}$ on ${displaystyle S^{n}}$ gives the structure of ${displaystyle O(1)}$-Main work on ${displaystyle mathbb {R} mathbb {P} ^{n}}$. Likewise, ${displaystyle S^{2n+1}$ It's a ${displaystyle U(1)}$-Main work on ${displaystyle mathbb {C} mathbb {P} ^{n}}$ and ${displaystyle S^{4n+3}$ It's a ${displaystyle Sp(1)}$-Main work on the quaternionic projective space ${displaystyle mathbb {H} mathbb {P} ^{n}}$. Then we have a series of main fibers for each positive integer ${displaystyle n}$:

${displaystyle {O}(1)to S(mathbb {R} ^{n+1})to mathbb {RP} ^{n}}$

${displaystyle {U}(1)to S(mathbb {C} ^{n+1})to mathbb {CP} ^{n}}$

${displaystyle {Sp}(1)to S(mathbb {H} ^{n+1})to mathbb {HP} ^{n}}}$

Here. ${displaystyle S(V)}$ denotes the sphere unity in ${displaystyle V}$. For all these examples the case ${displaystyle n=1}$ give Hopf's fibers.

Trivializations and sections

One of the most important questions regarding a bundle space is whether or not it is a trivial bundle (ie isomorphic to a product bundle). For the main bundles there is a convenient characterization of triviality:

Theorem. A main fiber is trivial if and only if you support a global section.

This result is not true for bundled individuals in general. In particular, vector bundles, for example, always have a zero section, whether they are trivial or not.

The same theorem applies to the local trivializations of major fiberdes. Sea ${displaystyle pi:Pto X}$ One ${displaystyle G}$- Main man. An open set ${displaystyle U}$ in ${displaystyle X}$ supports a local trivialization if and only if a local section exists ${displaystyle U}$. Given a local trivialization ${displaystyle phi:pi ^{-1}(U)to Utimes G}$ we can define an associated local section

${displaystyles:Uto pi ^{-1}(U)}$,

${displaystyle s(x):=phi ^{-1}(x,e)}$

where ${displaystyle e}$ is the identity in ${displaystyle G}$. reciprocally, given a local section ${displaystyle s}$ We can. define a trivialization ${displaystyle phi }$ for

${displaystyle phi ^{-1}(x,g)=s(x)cdot g}$

The fact that ${displaystyle G}$ acts in a simple and transient way ensures that this application is a bijection. It is possible to check which is also a homeomorphism. Local trivializations defined by a local section are ${displaystyle G}$-equivariants in the following sense: if we write

${displaystyle phi:pi ^{-1}(U)to Utimes G}$

in the form

${displaystyle phi (p)=(pi (p),varphi (p)}$

then the application ${displaystyle varphi:Pto G}$ sat

${displaystyle varphi (pcdot g)=varphi (p)cdot g}$

In terms of local sections ${displaystyle s}$, implementation ${displaystyle varphi }$ It's given by

${displaystyle varphi (s(x)cdot g)=g. !$

The local version of the section theorem then states that the equivalent local trivializations of a main bundle correspond to local sections.

Sea ${displaystyle ({U_{i},{phi _{i}}}}}$ a local trivialization equivariate of ${displaystyle P}$ and ${displaystyle s_{i}}$ local sections induced in each ${displaystyle U_{i}}$. In ${displaystyle U_{i}cap U_{j}}$ sections ${displaystyle s_{i}}$ and ${displaystyle s_{j}}$ are related by the group ${displaystyle G}$. Indeed, the transitional functions between the different trivializations, given by

${displaystyle (phi _{j}circ phi _{i}^{-1}): (U_{i}cap U_{j})times Gto (U_{i}cap U_{j})times G}$

in the first coordinate turn out to be the identity and send

${displaystyle (x,e)mapsto (x,t_{ij}(x)}$.

Then for any ${displaystyle xin U_{i}cap U_{j}}$ We've got

${displaystyle s_{j}(x)=s_{i}(x)cdot t_{ij}(x). !$

Characterization of differentiable main bundles

Yeah. ${displaystyle pi:Pto X}$ It's a ${displaystyle G}$- differential main fibre, then ${displaystyle G}$ acts in its own and free form in ${displaystyle P}$ so that orbit space ${displaystyle P/G}$is dimorphous to the base space ${displaystyle X}$. It turns out that this completely characterizes the main differential fibers. This is, if ${displaystyle P}$ is a differentiable variety, ${displaystyle G}$ It's a group of Lie and ${displaystyle mu:Ptimes Gto P}$ an action on the right differential, free and own then

• ${displaystyle P/G}$ (quoting space for action ${displaystyle mu }$) is a differentiable variety,
• the natural projection ${displaystyle pi:Pto P/G}$ It's a submersion, and
• ${displaystyle P}$ It's a ${displaystyle G}$- differential main fibre over ${displaystyle P/G}$.

Structural group reduction

Sea ${displaystyle pi:Pto X}$ It's a ${displaystyle G}$- Main man. Given a subgroup ${displaystyle Hsubset G}$, we can consider the fiber ${displaystyle P/H}$ whose fibers are the conjunction ${displaystyle G/H}$. If the new fiber supports a global section, we will say that the section is a reduction of structural group ${displaystyle G}$ to ${displaystyle H}$. In particular, if ${displaystyle H}$ is identity, then a reduction of ${displaystyle G}$ a identity is equivalent to having a global section of the original fiber, which is equivalent to trivial fiber. There are generally no reductions in the structural group.

Many questions about the topological structure of a bundle can be rephrased as questions about admissibility of the reduction of the structural group. For example:

• A real variety ${displaystyle 2n}$-dimensional supports a complex structure if the fiber of frames corresponding to the variety, whose fibers are ${displaystyle GL(2n,mathbb {R}}}$can be reduced to the group ${displaystyle GL(n,mathbb {C})subset GL(2n,mathbb {R}). !$
• A variety ${displaystyle n}$- dimensional admits ${displaystyle n}$ linearly independent vector fields at each point if your frame fiber is parallelizable, i.e. if the frame fiber supports a global section.
• A real variety ${displaystyle n}$- dimensional supports a field ${displaystyle k}$- I plan if the frame fiber can be reduced to the structural group ${displaystyle GL(k,mathbb {R})subset GL(n,mathbb {R}}}$