group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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 $GL(n)$ of the principal bundle to which the tangent bundle is associated along the given inclusion $G \hookrightarrow GL(n)$.
More generally, one can consider the case $G$ is not a subgroup but equipped with any group homomorphism $G \to GL(n)$. 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.
Given a smooth manifold $X$ of dimension $n$ and given a Lie subgroup $G \hookrightarrow GL(n)$ of the general linear group, then a $G$-structure on $X$ is a reduction of the structure group of the frame bundle of $X$ to $G$.
There are many more explicit (and more abstract) equivalent ways to say this, which we discuss below. There are also many evident variants and generalizations.
Notably one may consider reductions of the frames in the $k$th order jet bundle. (e. g. Alekseevskii) This yields order $k$ $G$-structure and the ordinary $G$-structures above are then first order.
Moreover, the definition makes sense for generalized manifolds modeled on other base spaces than just Cartesian spaces. In particular there is an evident generalization to supermanifolds.
A $(B,f)$-structure is
for each $n\in \mathbb{N}$ a pointed CW-complex $B_n \in Top_{CW}^{\ast/}$
equipped with a pointed Serre fibration
to the classifying space $B O(n)$ (def.);
for all $n_1 \leq n_2$ a pointed continuous function
$\iota_{n_1, n_2} \;\colon\; B_{n_1} \longrightarrow B_{n_2}$
which is the identity for $n_1 = n_2$;
such that for all $n_1 \leq n_2 \in \mathbb{N}$ these squares commute
where the bottom map is the canonical one (def.).
The $(B,f)$-structure is multiplicative if it is moreover equipped with a system of maps $\mu_{n_1,n_2} \colon B_{n_1}\times B_{n_2} \to B_{n_1 + n_2}$ which cover the canonical multiplication maps (def.)
and which satisfy the evident associativity and unitality, for $B_0 = \ast$ the unit, and, finally, which commute with the maps $\iota$ in that all $n_1,n_2, n_3 \in \mathbb{N}$ these squares commute:
and
Similarly, an $S^2$-$(B,f)$-structure is a compatible system
indexed only on the even natural numbers.
Generally, an $S^k$-$(B,f)$-structure for $k \in \mathbb{N}$, $k \geq 1$ is a compatible system
for all $n \in \mathbb{N}$, hence for all $k n \in k \mathbb{N}$.
(Lashof 63, Stong 68, beginning of chapter II, Kochmann 96, section 1.4)
See also at B-bordism.
Examples of $(B,f)$-structures (def. 1) include the following:
$B_n = B O(n)$ and $f_n = id$ is orthogonal structure (or “no structure”);
$B_n = E O(n)$ and $f_n$ the universal principal bundle-projection is framing-structure;
$B_n = B SO(n) = E O(n)/SO(n)$ the classifying space of the special orthogonal group and $f_n$ the canonical projection is orientation structure;
$B_n = B Spin(n) = E O(n)/Spin(n)$ the classifying space of the spin group and $f_n$ the canonical projection is spin structure.
Examples of $S^2$-$(B,f)$-structures include
Given a smooth manifold $X$ of dimension $n$, and given a $(B,f)$-structure as in def. 1, then a $(B,f)$-structure on the manifold is an equivalence class of the following structure:
an embedding $i_X \; \colon \; X \hookrightarrow \mathbb{R}^k$ for some $k \in \mathbb{N}$;
a homotopy class of a lift $\hat g$ of the classifying map $g$ of the tangent bundle
The equivalence relation on such structures is to be that generated by the relation $((i_{X})_1, \hat g_1) \sim ((i_{X})_,\hat g_2)$ if
$k_2 \geq k_1$
the second inclusion factors through the first as
the lift of the classifying map factors accordingly (as homotopy classes)
Given a smooth manifold $X$ of dimension $n$ with frame bundle $Fr(X)$, and given a Lie group monomorphism
into the general linear group, then a $G$-structure on $X$ is an $G$-principal bundle $P \to X$ equipped with an inclusion of fiber bundles
which is $G$-equivariant.
(Sternberg 64, section VII, def. 2.1).
From this perspective, a $G$-structure consists of the collection of all $G$-frames on a manifold. For instance for an orthogonal structure it consists of all those frames which are pointwise an orthonormal basis of the tangent bundle (with respect to the Riemannian metric which is defined by the orthonormal structure).
Accordingly:
Given $G \hookrightarrow GL(n)$ and given any one frame field $\sigma \colon X \to Fr(X)$ over a manifold $X$, then acting with $G$ on $\sigma$ at each point produces a $G$-subbundle. This is called the $G$-structure generated by the frame field $\sigma$.
A $G$-structure equipped with compatible connection data is equivalently a Cartan connection for the inclusion $(G \hookrightarrow \mathbb{R}^n \rtimes G)$.
See at Cartan connection – Examples – G-structures
We give an equivalent definition of $G$-structures in terms of higher differential 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.
Let $G \to K$ be a homomorphism of Lie groups. Write
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
be any choice of morphism modulating it.
Write $\mathbf{H}(X, \mathbf{B}G)$ 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
(the groupoid of twisted c-structures).
If here $k$ is trivial in that it factors through the point, $k \colon X \to \ast \to \mathbf{B}K$ then this homotopy fiber product is $\mathbf{H}(X,K/G)$, where $K/G$ is the coset space (Klein geometry) which itself sits in the homotopy fiber sequence
Specifically, when $X$ is a smooth manifold of dimension $n$, the frame bundle $Fr(X)$ is modulated? by a morphism $\tau_X \colon X \to \mathbf{B} GL(n)$ into the moduli stack for the general linear group $K := GL(n)$. Then for any group homomorphism $G \to GL(n)$, a $G$-structure on $X$ is a $G$-structure on $Fr(X)$, as above.
We discuss the concept in the generality of higher differential geometry, formalized in differential cohesion.
See at differential cohesion – G-Structure
A $G$-structure on a manifold $X$ is called locally flat (Sternberg 64, section VII, def. 24) or integrable (e.g. Alekseevskii) if it is locally equivalent to the standard flat $G$-structure, def. 7.
This means that there is an open cover $\{U_i \to X\}$ by open subsets of the Cartesian space $\mathbb{R}^n$ such that the restriction of the $G$-structure to each of these is equivalent to the standard flat $G$-structure.
See at integrability of G-structures for more on this
The obstruction to integrability of $G$-structure is the torsion of a G-structure. See there for more.
The existence of $G$-structures on tangent bundles of Riemannian manifolds is closely related to these having special holonomy.
Let $(X,g)$ be a connected Riemannian manifold of dimension $n$ with holonomy group $Hol(g) \subset O(n)$.
For $G \subset O(n)$ some other subgroup, $(X,g)$ admits a torsion-free G-structure precisely if $Hol(g)$ 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 $Hol(g)$ with them lands in $G$.
This appears as (Joyce prop. 3.1.8)
For $G \hookrightarrow GL(n)$ a subgroup, then the standard flat $G$-structure on the Cartesian space $\mathbb{R}^n$ is the $G$-structure which is generated, via def. 4, from the canonical frame field on $\mathbb{R}^n$ (the one which is the identity at each point, under the defining identifications).
For the subgroup of $GL(n, \mathbb{R})$ of matrices of positive determinant, a $GL(n, \mathbb{R})^+$-structure defines an orientation.
For the orthogonal group, an $O(n)$-structure defines a Riemannian metric. (See the discussion at vielbein and at
For the special linear group, an $SL(n,R)$-structure defines a volume form.
For the trivial group, an $\{e\}$-structure consists of an absolute parallelism? of the manifold.
For $n = 2 m$ even, a $GL(m, \mathbb{C})$-structure defines an almost complex structure on the manifold. It must satisfy an integrability condition to be a complex structure.
An example for a lift of structure groups is
This continues with lifts to the
string group giving string structure;
fivebrane group giving fivebrane structure.
For general $G \to K$, the corresponding notion of Cartan geometry involves $G$-structure on $K$-principal bundles (not necessarily underlying a tangent bundle).
A $U(n,n) \hookrightarrow O(2n,2n)$-structure is a generalized complex structure;
For $H_n \to E_{n(n)}$ the inclusion of the maximal compact subgroup into the split real form of an exceptional Lie group, the corresponding structure is an exceptional generalized geometry.
See the list at tiwsted differential c-structure.
Need to talk about integrability conditions, and those of higher degree. Also need to discuss pseudo-groups?.
The concept of topological $G$-structure (lifts of homotopy classes of classifying maps) originates with cobordism theory. Early expositions in terms of (B,f)-structures include
Richard Lashof, Poincaré duality and cobordism, Trans. AMS 109 (1963), 257-277
Robert Stong, beginning of chapter II of Notes on Cobordism theory, 1968 (toc pdf, publisher page)
Stanley Kochmann, section 1.4 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
The concept in differential geometry originates around the work of Eli Cartan (Cartan geometry) and
Textbook accounts include
Shlomo Sternberg, chapter VII of Lectures on differential geometry, Prentice-Hall (1964)
Shoshichi Kobayashi, Katsumi Nomizu, Foundations of differential geometry , Volume 1 (1963), Volume 2 (1969), Interscience Publishers, reprinted 1996 by Wiley Classics Library
Surveys include
D. V: Alekseevskii, $G$-structure on a manifold in M. Hazewinkel (ed.) Encyclopedia of Mathematics, Volume 4
Discussion with an eye towards special holonomy is in
Discussion with an eye towards torsion constraints in supergravity is in
Discussion of $G$-structures in supergeometry includes
Some discussion is in section 4.4.2 of