Reduction of the structure group
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In [[mathematics]], in particular the theory of [[principal bundle]]s, one can ask if a <math>G</math>-bundle "comes from" a subgroup <math>H < G</math>. This is called '''reduction of the structure group''' (to <math>H</math>), and makes sense for any map <math>H \to G</math>, which need not be an inclusion (despite the terminology).
==Definition==
Formally, given a ''G''-bundle ''B'' and a map ''H'' → ''G'' (which need not be an inclusion),
a '''reduction of the structure group''' (from ''G'' to ''H'') is an ''H''-bundle <math>B_H</math> such that the [[pushout (category theory)|pushout]] <math>B_H \times_H G</math> is isomorphic to ''B''.
Note that these do not always exist, nor if they exist are they unique.
As a concrete example, every even dimensional real vector space is the underlying real space of a complex vector space: it admits a [[linear complex structure]]. A real vector bundle admits an [[almost complex]] structure if and only if it is the underlying real bundle of a complex vector bundle. This is a reduction along the inclusion ''GL''(''n'','''C''') → ''GL''(2''n'','''R''')
In terms of transition maps, a ''G''-bundle can be reduced if and only if the transition maps can be taken to have values in ''H''.
Note that the term ''reduction'' is misleading: it suggests that ''H'' is a subgroup of ''G'', which is often the case, but need not be (for example for [[spin structure]]s): it's properly called a [[Homotopy lifting property|lifting]].
More abstractly, "''G''-bundles over ''X''" is a [[functor]]<ref>Indeed, it is a [[bifunctor]] in ''G'' and ''X''.</ref> in ''G'': given a map ''H'' → ''G'', one gets a map from ''H''-bundles to ''G''-bundles by [[Induced representation|inducing]] (as above). Reduction of the structure group of a ''G''-bundle ''B'' is choosing an ''H''-bundle whose image is ''B''.
The inducing map from ''H''-bundles to ''G''-bundles is in general neither onto nor one-to-one, so the structure group cannot always be reduced, and when it can, this reduction need not be unique. For example, not every manifold is [[orientable]], and those that are orientable admit exactly two orientations.
If ''H'' is a Lie subgroup of ''G'', then there is a natural one-to-one correspondence between reductions of a ''G''-bundle ''B'' to ''H'' and global sections of the [[fiber bundle]] ''B''/''H'' obtained by quotienting ''B'' by the right action of ''H''. Specifically, the fibration ''B'' → ''B''/''H'' is a principal ''H''-bundle over ''B''/''H''. If σ : ''X'' → ''B''/''H'' is a section, then the [[pullback bundle]] ''B''<sub>H</sub> = σ<sup>-1</sup>''B'' is a reduction of ''B''.
==Examples==
Examples for [[vector bundle]]s, particularly the [[tangent bundle]] of a [[manifold]]:
* <math>GL^+ < GL</math> is an [[orientation]], and this is possible if and only if the bundle is orientable
* <math>SL < GL</math> is a [[volume form]]; since <math>SL \to GL^+</math> is a [[deformation retract]], a volume form exists if and only if a bundle is orientable
* <math>SL^{\pm} < GL</math> is a pseudo-[[volume form]], and this is always possible
* <math>O(n) < GL(n)</math> is a metric; as <math>O(n)</math> is the [[maximal compact subgroup]] (so the inclusion is a deformation retract), this is always possible
* <math>GL(n,\mathbf{C}) < GL(2n,\mathbf{R})</math> is an [[almost complex manifold|almost complex structure]]
* <math>\mbox{Spin}(n) \to \mbox{SO}(n)</math> (which is ''not'' an inclusion: it's a 2-fold [[covering space]]) is a [[spin manifold|spin structure]].
* <math>GL(k) \times GL(n-k) < GL(n)</math> decomposes a vector bundle as a [[Whitney sum]] (direct sum) of sub-bundles of rank ''k'' and ''n'' − ''k''.
==Integrability==
Many geometric structures are stronger than ''G''-structures; they are ''G''-structures with an ''integrability condition''. Thus such a structure requires a reduction of the structure group (and can be obstructed, as below), but this is not sufficient. Examples include [[complex manifold|complex structure]], [[symplectic manifold|symplectic structure]] (as opposed to [[almost complex structure]]s and [[almost symplectic structure]]s).
Another example is for a [[foliation]], which requires a reduction of the [[tangent bundle]] to a block matrix subgroup, together with an integrability condition so that the [[Frobenius theorem (differential topology)|Frobenius theorem]] applies.
==Obstruction==
''G''-bundles are classified by the [[classifying space]] ''BG'', and similarly ''H''-bundles are classified by the classifying space ''BH'', and the induced ''G''-structure on an ''H''-bundle corresponds to the induced map <math>BH \to BG</math>. Thus given a ''G''-bundle with classifying map <math>\xi\colon X \to BG</math>, the obstruction to the reduction of the structure group is the class of <math>\xi</math> as a map to the [[cofiber]] <math>BG/BH</math>; the structure group can be reduced if and only if the class of <math>\bar \xi</math> is [[null-homotopic]].
When <math>H \to G</math> is a [[homotopy equivalence]], the cofiber is contractible, so there is no obstruction to reducing the structure group, for example for <math>O(n) \to GL(n)</math>.
Conversely, the cofiber induced by the inclusion of the trivial group <math>e \to G</math> is again <math>BG</math>, so the obstruction to an [[absolute parallelism]] (trivialization of the bundle) is the class of the bundle.
===Structure over a point===
As a simple example, there is no obstruction to reducing the structure group of a <math>G</math>-''space'' to an <math>H</math>-''space'', thinking of a <math>G</math>-space as a <math>G</math>-bundle over a point, as in that case the classifying map is [[null-homotopic]], as the domain is a point. Thus there is no obstruction to "reducing the structure group" of a vector space: thus every vector space admits an orientation, and so forth.
==See also==
* [[associated bundle]]
==References==
<references/>
[[Category:Structures on manifolds]]
[[Category:Fiber bundles]]
[[Category:Differential topology]]
[[Category:Differential geometry]]