nLab principal SO(3)-bundle

Idea

Principal SO(3)-bundles are principal bundles with the third special orthogonal group $SO(3)$ as structure group/gauge group. Applications include the frame bundle of an orientable 3-manifold as well as the (anti) self-dual bundles $\Lambda_\pm^2T X$ for an orientable Riemannian 4-manifold $X$, resulting from the splitting $\mathfrak{so}(4)\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)$.

Principal $SO(3)$-bundles are in particular induced by principal SU(2)-bundles using the double cover $SU(2)\cong Spin(3)\twoheadrightarrow SO(3)$ and principal U(2)-bundles using the canonical projection $U(2)\cong Spin^\mathrm{c}(3)\cong Spin^\mathrm{h}(2)\rightarrow SO(3)$. In fact the latter example generalizes to every $Spin^\mathrm{h}(n)$-bundle using the double cover of the former example:

$$Spin^\mathrm{h}(n) \cong(Spin(n)\times SU(2))/\mathbb{Z}_2 \rightarrow SU(2)/\mathbb{Z}_2 \cong SO(3).$$

Principal SO(3)-bundles also induce principal SO(2)-bundles and are induced by principal SO(4)-bundles and principal SO(6)-bundles using the canonical inclusions $SO(2)\hookrightarrow SO(3)\hookrightarrow SO(4)\hookrightarrow SO(6)$.

Principal SO(3)-bundles also arise from any principal $G$-bundle with a three-dimensional real Lie group $G$ using its adjoint representation $Ad\colon G\rightarrow SL_3(\mathbb{R})$, which induces a map $\mathcal{B}Ad\colon\mathcal{B}G\rightarrow B SO(3)$.

Characteristic classes

Proposition

A principal SO(3)-bundle $P$ fulfills:

$$w_2^2(P) \equiv p_1(P) \mod 2.$$

(In general, a principal $SO(n)$-bundle $P$ fulfills $w_{2k}^2(P)\equiv p_k(P) \mod 2$ for $2k\leq n$.)

(Milnor & Stasheff 74, Prob. 15-A, Gompf & Stipsicz 99, Ex. 1.4.21 d, Hatcher 17, Prop. 3.15 a)

Proposition

A principal SO(3)-bundle $P$ fulfills:

$$w_3(P) \equiv e(P) \mod 2.$$

(In general, a principal $SO(n)$-bundle $P$ fulfills $w_n(P)\equiv e(P) \mod 2$.)

(Milnor & Stasheff 74, Prop. 9.5, Hatcher 17, Prop. 3.13 c)

Proposition

A principal SO(3)-bundle $P$ fulfills:

$$e(P) =W_3(P) =\beta w_2(P).$$

with the third integral Stiefel-Whitney class and the Bockstein homomorphism $\beta\colon H^2(M,\mathbb{Z}_2)\rightarrow H^3(M,\mathbb{Z})$ of the short exact sequence $0\rightarrow\mathbb{Z}\hookrightarrow\mathbb{Z}\twoheadrightarrow\mathbb{Z}_2\rightarrow 0$. (In general, a principal $SO(2n+1)$-bundle $P$ fulfills $e(P)=W_{2n+1}(P)=\beta w_{2n}(P)$.)

(Milnor & Stasheff 74, Prob. 15-D, Hatcher 17, Ch. 3, Ex. 3)

Classification over 4-manifolds by characteristic classes

Proposition

Let $X$ be a 4-manifold. Two principal SO(3)-bundles $P,Q\twoheadrightarrow X$ are isomorphic if and only if their second Stiefel-Whitney class and first Pontrjagin class are equal:

$$w_2(P) =w_2(Q) \in H^2(X,\mathbb{Z}_2);$$

$$p_1(P) =p_1(Q) \in H^4(X,\mathbb{Z}).$$

(But the classes are not independent from each other according to the previous proposition.)

(Dold & Whitney 59, Thrm. 1 a, Gompf & Stipsicz 99, Thrm. 1.4.20)

Certain 4-manifolds yield simplifcations when the singular cohomology groups vanish. For example, for principal SO(3)-bundles over the 4-sphere the second singular cohomology vanishes and therefore the condition of equal second Stiefel-Whitney classes becomes trivial. In this case one has a group isomorphism:

$$p_1\colon \pi_4 B SO(3) \cong\pi_3 SO(3) \xrightarrow\cong\mathbb{Z}.$$

An abstract way to obtain this group isomorphism is the exceptional isomorphism $Spin(3)\cong SU(2)\cong S^3$, which double covers $SO(3)$ and therefore has the same higher homotopy groups (beyond the fundamental group). Therefore $\pi_3 SO(3)\cong\pi_3(S^3)\cong\mathbb{Z}$. (Just the result is also stated in Dold & Whitney 59, Eq. (2).)

Proposition

Let $X$ be a $4$-manifold. For every $(w_2,p_1)\in H^2(X,\mathbb{Z}_2)\times H^4(X,\mathbb{Z})$ with $p_1=Pont(w_2) \mod 4$, there exists a principal SO(3)-bundle $P\twoheadrightarrow X$ with $w_2(P)=w_2$ and $p_1(P)=p_1$.

(Gompf & Stipsicz 99, Thrm. 1.4.20)

Liftings

Proposition

A principal SO(3)-bundle $f\colon X\rightarrow B SO(3)$ lifts to a principal SU(2)-bundle $\widehat{f}\colon X\rightarrow B SU(2)\cong\mathbb{H}P^\infty$ if and only if its second Stiefel-Whitney class vanishes, hence the composition $w_2\circ f\colon X\rightarrow K(\mathbb{Z}_2,2)$ is nullhomotopic.

Let $P$ be a principal $SU(2)$-bundle and $Q$ be the associated principal $SO(3)$-bundle. One then has:

$$p_1(Q) =p_1 Ad(P) =-4c_2(P);$$

$$w_2(Q) =w_2Ad(P) =0.$$

(Uhlenbeck & Freed 91, Appendix E.6)

Proposition

A principal SO(3)-bundle $f\colon X\rightarrow B SO(3)$ lifts to a principal U(2)-bundle $\widehat{f}\colon X\rightarrow B U(2)$ if and only if its third integral Stiefel-Whitney class vanishes, hence the composition $W_3\circ f\colon X\rightarrow K(\mathbb{Z},2)$ is nullhomotopic. If the second Stiefel-Whitney class $w_2\circ f\colon X\rightarrow K(\mathbb{Z}_2,2)$ lifts to an integral class $c\colon X\rightarrow K(\mathbb{Z},2)$, then it is exactly the first Chern class of the lift with $c_1\circ\widehat{f}=c$.

(Donaldson & Kronheimer 91, p. 42)

In the situation of this lemma, it is often more useful to consider the exceptional isomorphism $SO(3)\cong PU(2)$ to the projective unitary group.

(Anti) self-dual bundles

Using the exceptional isomorphisms $Spin(3)\cong SU(2)$ and $Spin(4)\cong SU(2)\times SU(2)$, matrices in SO(3) and SO(4) can be represented by matrices in SU(2). In particular there are maps:

$$\phi\colon SO(3)\rightarrow SO(4) [U]\mapsto[(U,U)];$$

$$\psi_-\colon SO(4)\rightarrow SO(3), [(U_-,U_+)]\mapsto[U_-];$$

$$\psi_+\colon SO(4)\rightarrow SO(3), [(U_-,U_+)]\mapsto[U_+]$$

with $\psi_\pm\circ\phi=id$ and so that there is a Lie algebra isomorphism:

$$(\psi_-',\psi_+')\colon \mathfrak{so}(4)\xrightarrow\cong\mathfrak{so}(3)\oplus\mathfrak{so}(3)$$

Proposition

For an orientable Riemannian 4-manifold $X$, one has $\Lambda_\pm^2T X=(\psi_\pm)_*T X$ with characteristic classes:

$$w_2(\Lambda_+^2T X) =w_2(X);$$

$$p_1(\Lambda_+^2T X) =p_1(X) +2e(X).$$

(Gompf & Stipsicz 99, Thrm. 1.4.20)

According to the first equation, $\Lambda_+^2T X$ lifts to a principal SU(2)-bundle if and only if $T X$ lifts to a pair of principal SU(2)-bundles (being $Spin(3)$ and $Spin(4)$ structures repsectively). Or just taking the weaker implied equality of the third integral Stiefel-Whitney classes, $\Lambda_+^2T X$ lifts to a principal U(2)-bundle if and only if $T X$ lifts to a pair of principal U(2)-bundles with same first Chern class (being $Spin^\mathrm{c}(3)$ and $Spin^\mathrm{c}(4)$ structures respectively).

According to the second equation, using the fundamental class $[X]\in H_4(X,\mathbb{Z})\cong\mathbb{Z}$ induced by the orientation as well as Hirzebruch's signature theorem, one has:

$$\big\langle p_1(\Lambda_+^2T X),[X]\big\rangle =\big\langle p_1(X),[X]\big\rangle +2\big\langle e(X),[X]\big\rangle =3\sigma(X) +2\chi(X)$$

This expression appears both in the virtual dimension of the Seiberg-Witten moduli space (Gompf & Stipsicz 99, Thrm. 2.4.24) and Noether’s formula for almost complex structures. (Gompf & Stipsicz 99, Thrm. 1.4.15)

Examples

• One has $S^n\cong SO(n+1)/SO(n)$, hence there is a principal SO(3)-bundle $SO(4)\twoheadrightarrow S^3$. Such principal bundles are classified by:
  $$\pi_3B SO(3) \cong\pi_2 SO(3) \cong 1.$$

  Hence the bundle is trivial and $SO(4)\cong SO(3)\times S^7$.

Application in Yang-Mills theory

Proposition

Reducible anti self-dual Yang-Mills connections of a principal $SO(3)$-bundle $E$ over a compact, simply connected, oriented Riemannian 4-manifold $X$ are in bijective correspondence with non-zero pairs $\pm c\in H^2(X,\mathbb{Z})$ with $c^2=p_1(E)$.

(Donaldson & Kronheimer 97, Prop. (4.2.15))

Proposition

Irreducible anti self-dual Yang-Mills connections of a principal $SO(3)$-bundle $E$ over a simply connected Riemannian 4-manifold $X$ are still irreducible under the restriction to any non-empty open subset.

(Donaldson & Kronheimer 97, Lem. (4.3.21))

Related concepts

Particular principal bundles:

• double cover (principal O(1)-bundle)
• principal O(2)-bundle
• principal U(1)-bundle (principal SO(2)-bundle)
• principal U(2)-bundle (principal $Spin^\mathrm{c}(3)$-bundle, principal $Spin^\mathrm{h}(2)$-bundle)
• principal U(3)-bundle
• principal SO(3)-bundle
• principal SO(4)-bundle ($Spin^\mathrm{h}(3)$-bundle)
• principal SO(5)-bundle
• principal SO(6)-bundle
• principal SO(7)-bundle
• principal SO(8)-bundle
• principal SU(2)-bundle (principal Sp(1)-bundle, principal Spin(3)-bundle)
• principal SU(3)-bundle
• principal SU(4)-bundle (principal Spin(6)-bundle)
• principal Sp(2)-bundle (principal Spin(5)-bundle)

References

• Albrecht Dold and Hassler Whitney, Classification of oriented sphere bundles over a 4-complex (1959), Annals of Mathematics Vol. 69 No. 3 [doi:10.2307/1970030 ]

• Friedrich Hirzebruch, Heinz Hopf, Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten (1958)

• Albrecht Dold and Hassler Whitney, Classification of oriented sphere bundles over a 4-complex (1959)

• John Milnor, Jim Stasheff, Characteristic classes, Princeton Univ. Press (1974) (ISBN:9780691081229, doi:10.1515/9781400881826, pdf)

• Daniel Freed, Karen Uhlenbeck, Instantons and Four-Manifolds, Mathematical Sciences Research Institute Publications, Springer 1991 ([doi:10.1007/978-1-4613-9703-8]

  (https://link.springer.com/book/10.1007/978-1-4613-9703-8))

• Simon Donaldson, Peter Kronheimer: The Geometry of Four-Manifolds (1990, revised 1997), Oxford University Press and Claredon Press, [oup:52942, doi:10.1093/oso/9780198535539.001.0001, ISBN:978-0198502692, ISSN:0964-9174]

• Robert Gompf and András Stipsicz, 4-Manifolds and Kirby Calculus (1999), Graduate Studies

  in Mathematics, Volume 20 [ISBN: 978-0-8218-0994-5, doi:10.1090/gsm/020]

• Allen Hatcher: Vector bundles and K-Theory, book draft (2017) [webpage, pdf]

• Andrew Lobb, The Dold-Whitney theorem and the Sato-Levine invariant (2017), [arxiv:1709.09922 pdf]