cohomology

# Contents

## Idea

A G-structure on an $n$-manifold $M$, for a given structure group $G$, is a $G$-subbundle of the frame bundle (of the tangent bundle) of $M$.

Equivalently, this means that a $G$-structure is a choice of reduction of the canonical structure group $\mathrm{GL}\left(n\right)$ of the principal bundle to which the tangent bundle is associated along the given inclusion $G↪\mathrm{GL}\left(n\right)$.

More generally, one can consider the case $G$ is not a subgroup but equipped with any group homomorphism $G\to \mathrm{GL}\left(n\right)$. If this is instead an epimorphism one speaks of a lift of structure groups.

Both cases, in turn, can naturally be understood as special cases of twisted differential c-structures, which is a notion that applies more generally to principal infinity-bundles.

## Definition

(…)

### In terms of higher geometry

We give an equivalent definition of $G$-structures in terms of higher geometry (“from the nPOV”). This serves to clarify the slightly subtle but important difference between existence and choice of $G$-structure, and seamlessly embeds the notion into the more general context of twisted differential c-structures.

###### Definition

Let $G\to K$ be a homomorphism of Lie groups. Write

$c:BG\to BK$\mathbf{c} : \mathbf{B}G \to \mathbf{B}K

for the morphism of delooping Lie groupoids ( the smooth moduli stacks of smooth $K$- and $G$-principal bundles, respectively).

For $X$ a smooth manifold (or generally an orbifold or Lie groupoid, etc.) Let $P\to X$ be a $K$-principal bundle and let $g:X\to BK$ be any choice of morphism representing it.

Write $H\left(X,BG\right)$ etc. for the hom-groupoid of smooth groupoids / smooth stacks . This is equivalently the groupoid of $G$-principal bundles over $X$ and smooth gauge transformations between them.

Then the groupoid of $G$-structure on $P$ (with respect to the given morphism $G\to K$) is the homotopy pullback

$c{\mathrm{Struc}}_{\left[P\right]}\left(X\right):=H\left(X,BG\right){×}_{H\left(X,BK\right)}\left\{P\right\}\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{c}Struc_{[P]}(X) := \mathbf{H}(X, \mathbf{B}G) \times_{\mathbf{H}(X, \mathbf{B}K)} \{P\} \,.
$\begin{array}{ccc}c{\mathrm{Struc}}_{\left[P\right]}\left(X\right)& \to & *\\ ↓& {⇙}_{\simeq }& {↓}^{g}\\ H\left(X,BG\right)& \stackrel{H\left(X,c\right)}{\to }& H\left(X,BK\right)\end{array}$\array{ \mathbf{c}Struc_{[P]}(X) &\to& * \\ \downarrow & \swArrow_\simeq& \downarrow^{g} \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{\mathbf{H}(X, \mathbf{c})}{\to}& \mathbf{H}(X, \mathbf{B}K) }

(the groupoid of twisted c-structures).

Specifically, when $X$ is a smooth manifold of dimension $n$, the tangent bundle $TX$ canonically comes from a morphism $TX:X\to B\mathrm{GL}\left(n\right)$ into the moduli stack for the general linear group $K:=\mathrm{GL}\left(n\right)$. Then for any group homomorphism $G\to \mathrm{GL}\left(n\right)$, a $G$-structure on $X$ is a $G$-structure on $TX$, as above.

### Integrability

For the moment see at integrability of G-structures.

## Properties

### Relation to special holonomy

The existence of $G$-structures on tangent bundles of Riemannian manifolds is closely related to these having special holonomy.

###### Theorem

Let $\left(X,g\right)$ be a connected Riemannian manifold of dimension $n$ with holonomy group $\mathrm{Hol}\left(g\right)\subset O\left(n\right)$.

For $G\subset O\left(n\right)$ some other subgroup, $\left(X,g\right)$ admits a torsion-free G-structure precisely if $\mathrm{Hol}\left(g\right)$ is conjugate to a subgroup of $G$.

Moreover, the space of such $G$-structures is the coset $G/L$, where $L$ is the group of elements suchthat conjugating $\mathrm{Hol}\left(g\right)$ with them lands in $G$.

This appears as (Joyce prop. 3.1.8)

## Examples

### Reduction of tangent bundle structure

• For the subgroup of $\mathrm{GL}\left(n,ℝ\right)$ of matrices of positive determinant, a $\mathrm{GL}\left(n,ℝ{\right)}^{+}$-structure defines an orientation.

• For the orthogonal group, an $O\left(n\right)$-structure defines a Riemannian metric. (See the discussion at vielbein and at

• For the special linear group, an $\mathrm{SL}\left(n,R\right)$-structure defines a volume form.

• For the trivial group, an $\left\{e\right\}$-structure consists of an absolute parallelism? of the manifold.

• For $n=2m$ even, a $\mathrm{GL}\left(m,ℂ\right)$-structure defines an almost complex structure on the manifold. It must satisfy an integrability condition to be a complex structure.

### Lift of tangent bundle structure

An example for a lift of structure groups is

• for the spin group $\mathrm{spin}\left(n\right)$, a $G$-structure is a spin structure.

This continues with lifts to the

### Higher geometric examples

See the list at tiwsted differential c-structure.

## Further issues

Need to talk about integrability conditions, and those of higher degree. Also need to discuss pseudo-groups?.

## References

Surveys include

Original articles include

The relation to special holonomy is discussed in

• Dominic Joyce, Compact manifolds with special holonomy , Oxford Mathematical Monogrophs (200)

General textbooks of differential geometry include

• Shoshichi Kobayashi, Katsumi Nomizu, Foundations of differential geometry

• Shlomo Sternberg, Lectures on differential geometry

### In higher geometry

Some discussion is in section 4.4.2 of

Revised on September 22, 2013 17:05:18 by Urs Schreiber (89.204.139.76)