nLab principal SO(4)-bundle

Idea

Principal SO(4)-bundles are special principal bundles with the fourth special orthogonal group $SO(4)$ as structure group/gauge group. Applications include the frame bundle of an orientable 4-manifold and John Milnor‘s construction of exotic 7-spheres.

Principal SO(4)-bundles in particular are induced by pairs of principal SU(2)-bundles using the canonical projection $SU(2)\times SU(2)\cong Spin(4)\twoheadrightarrow SO(4)$. Principal SO(4)-bundles also induce principal SO(2)-bundles and principal SO(3)-bundles and are induced by principal SO(6)-bundles using the canonical inclusions $SO(2)\hookrightarrow SO(3)\hookrightarrow SO(4)\hookrightarrow SO(6)$.

Principal SO(4)-bundles also arise from any principal $G$-bundle with a four-dimensional real Lie group $G$ using its adjoint representation $Ad\colon G\rightarrow SL_4(\mathbb{R})$, which induces a map $\mathcal{B}Ad\colon\mathcal{B}G\rightarrow B SO(4)$.

Characteristic classes

Proposition

A principal SO(4)-bundle $P$ fulfills:

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

$$w_4^2(P) \equiv p_2(P) \mod 2.$$

(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)

Let $X$ be an orientable 4-manifold (with fundamental class $[X]\in H^4(X,\mathbb{Z})\cong\mathbb{Z}$) and $P=Fr_{SO}TX$ be the frame bundle of its tangent bundle (which doesn’t change characteristic classes). Using Hirzebruch's signature theorem (Gompf & Stipsicz 99, Thrm. 1.4.12) connects one of itsa Stiefel-Whitney numbers with its signature:

$$w_2^2[X] =p_1[X] \mod 2 =\sigma(X) \mod 2.$$

Proposition

A principal SO(4)-bundle $P$ fulfills:

$$p_2(P) =e^2(P).$$

(In general, a principal $SO(2n)$-bundle $P$ fulfills $p_n(P)=e^2(P)$.)

(Milnor & Stasheff 74, Crl. 15.8, Hatcher 17, Prop. 3.15 b)

The two previous propositions together imply $w_4^2(P)\equiv e^2(P) mod 2$ and one even has:

Proposition

A principal SO(4)-bundle $P$ fulfills:

$$w_4(P) \equiv e(P) \mod 2.$$

(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)

Classification over 4-manifolds by characteristic classes

Proposition

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

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

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

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

(But the classes are not independent from each other according to a 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(4)-bundles over the 4-sphere, used in John Milnor‘s construction of exotic 7-spheres, 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:

$$\left(e,-\frac{1}{4}(p_1+2e)\right) \,\colon\, \pi_4 B SO(4) \cong\pi_3 SO(4) \xrightarrow\cong\mathbb{Z}\oplus\mathbb{Z}.$$

(Bais 24, Eq. 1)

An abstract way to obtain this group isomorphism is the exceptional isomorphism $Spin(4)\cong SU(2)\times SU(2)\cong S^3\times S^3$, which double covers $SO(4)$ and therefore has the same higher homotopy groups (beyond the fundamental group). Therefore $\pi_3 SO(4)\cong\pi_3(S^3\times S^3)\cong\mathbb{Z}\oplus\mathbb{Z}$. (Just the result is also stated in Dold & Whitney 59, Eq. (2).)

Theorem

(Dold-Whitney theorem) A closed orientable 4-manifold $X$ is parallelizable if and only if its its second Stiefel-Whitney class $w_2(X)\in H^2(X,\mathbb{Z})$, first Pontrjagin class $p_1(X)\in H^4(X,\mathbb{Z})$ and Euler class $e(X)\in H^4(X,\mathbb{Z})$ all vanish.

(Bais 24, Thrm. 1)

All three conditions can be interpreted geometrically. Let therefore $[X]\in H_4(X,\mathbb{Z})\cong\mathbb{Z}$ be the fundamental class induced by the orientation with the Kronecker pairing $\langle-,[X]\rangle\colon H^4(X,\mathbb{Z})\xrightarrow\cong\mathbb{Z}$ being a group isomorphism.

• $w_2(X)=0$ is equivalent to $X$ being a spin manifold. (Gompf & Stipsicz 99, Prop. 1.4.25)

• $p_1(X)=0$ is equivalent to a vanishing signature $\sigma(X)=0$ using Hirzebruch's signature theorem (Gompf & Stipsicz 99, Thrm. 1.4.12):

  $$\sigma(X) \;=\; \frac{1}{3}\big\langle p_1(X), [X] \big\rangle \mathrlap{\,.}$$

  Since the signature is a group isomorphism $\sigma\colon\Omega_4^SO\xrightarrow\cong\mathbb{Z}$, having $\sigma(X)=0$ is equivalent to the $X$ being the boundary of an orientable 5-manifold.

• $e(X)=0$ is equivalent to a vanishing Euler characteristic $\chi(X)=0$ using:

  $$\chi(X) \;=\; \big\langle e(X), [X] \big\rangle \mathrlap{\,.}$$

  Equivalently, $X$ has a nowhere vanishing vector field.

Liftings

Proposition

A principal SO(4)-bundle $f\colon X\rightarrow B SO(4)$ lifts to a pair of principal SU(2)-bundles $\widehat{f}_1,\widehat{f}_2\colon X\rightarrow B SU(2)$ 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.

Examples

• One has $S^n\cong SO(n+1)/SO(n)$, hence there is a principal SO(4)-bundle $SO(5)\twoheadrightarrow S^4$. Such principal bundles are classified by:
  $$\pi_4B SO(4) \cong\pi_3 SO(4) \cong\mathbb{Z}^2.$$

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 ]

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

• 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]

• Valentina Bais, On Dold-Whitney’s parallelizability of 4-manifolds (2024) [arxiv:2410.22117 ]