nLab spin^c



Higher spin geometry

Group Theory




By definition of the spin group Spin(n)Spin(n) there is a canonical inclusion

/2Spin(n) \mathbb{Z}/2\mathbb{Z}\hookrightarrow Spin(n)

of the group of order 2. For Spin(n)GL 1(Cl( n))Spin(n)\hookrightarrow GL_1(Cl(\mathbb{R}^n)) canonically realized by even Clifford algebra elements of unit norm, this is given by the inclusion of {+1,1}\{+1,-1\}.

We frequently write 2\mathbb{Z}_2 as shorthand for /2\mathbb{Z}/2\mathbb{Z}.


For nn \in \mathbb{N}, the Lie group Spin c(n)Spin^c(n) is the quotient group

Spin c(n) Spin(n)× 2U(1) =(Spin(n)×U(1))/ 2, \begin{aligned} Spin^c(n) & \coloneqq Spin(n) \times_{\mathbb{Z}_2} U(1) \\ & = (Spin(n) \times U(1))/{\mathbb{Z}_2} \,, \end{aligned}

of the product of the spin group with the circle group by the common sub-group of order 2 2Spin\mathbb{Z}_2 \hookrightarrow Spin and 2U(1)\mathbb{Z}_2 \hookrightarrow U(1) (i.e.: the central product group).

Usually the only the case n3n \geq 3 is considered.

Some authors (e.g. Gompf 97, p. 2) denote this as

Spin c(n) Spin(n)Spin(2) Spin(n)U(1) \begin{aligned} Spin^c(n) & \coloneqq Spin(n)\cdot Spin(2) \\ & \simeq Spin(n) \cdot U(1) \end{aligned}

following the notation Sp(n).Sp(1) (see there).


  • For n=3n=3, the exceptional isomorphism between Spin(3) and SU(2) extends to an isomorphism between Spin c(3)Spin^c(3) and the unitary group U(2)U(2):

    Spin c(3)U(2) Spin^c(3) \;\simeq\; U(2)

    over the exceptional isomorphism SO ( 3 ) PU ( 2 ) SO(3) \simeq PU(2) , as both of these quotient groups are quotients by the respective centers, both identifiable with the circle group U(1)U(1)) (Ozbagci–Stipsicz 2004 Section 6.2). This isomorphism follows from considering the surjective homomorphism SU ( 2 ) × U ( 1 ) U ( 2 ) SU(2) \times U(1) \to U(2) given by (A,z)zA(A,z) \mapsto z A, and noticing its kernel is precisely {±(I,1)}\{\pm(I,1)\}, hence U(2)U(2) satisfies the universal property defining Spin c(3)Spin^c(3) as a quotient.

  • For n=4n=4, we have

    Spin c(4)=(SU(2)×SU(2)×U(1))/{±(I,I,1)}U(2)× U(1)U(2). Spin^c(4) = \big(SU(2)\times SU(2)\times U(1)\big) / \{\pm(I,I,1)\} \simeq U(2)\times_{U(1)} U(2).

    This latter group is the fibre product of groups over U(1)U(1), namely {(A,B)U(2)×U(2)det(A)=det(B)}\{(A,B)\in U(2)\times U(2)\mid \det(A) = \det(B)\} (Ozbagci–Stipsicz 2004 Section 6.3). The construction can be seen by considering the surjective homomorphism

    SU(2)×SU(2)×U(1)SU(2)×SU(2)×U(1)×U(2)U(2)×U(2) SU(2) \times SU(2) \times U(1) \to SU(2) \times SU(2) \times U(1) \times U(2) \to U(2) \times U(2)

    defined by (A,B,z)(zA,zB)(A,B,z) \mapsto (z A,z B), which has image U(2)× U(1)U(2)U(2)\times_{U(1)} U(2). Similarly to the case of Spin c(3)Spin^c(3), the kernel consists of triples (A,B,z)(A,B,z) such that zA=I=zBz A = I = z B, hence that A=B=(1/z)IA = B = (1/z) I. Since det(A)=1/z 2=1\det(A) = 1/z^2 = 1, we must have z=±1z=\pm 1, and hence A=B=±IA=B=\pm I with the same sign as zz. Thus the kernel is precisely {±(I,I,1)}\{\pm(I,I,1)\}, and so again by the universal property we get the isomorphism as stated.

    The Spin ( 4 ) Spin(4) subgroup can be seen as the subgroup of pairs of unitary matrices with both of them having determinant 1.

  • Using the exceptional isomorphism Spin ( 6 ) SU ( 4 ) Spin(6) \simeq SU(4) , and the multiplication map SU(4)×U(1)U(4)SU(4)\times U(1) \to U(4) analogous to the above, it can be seen that the group Spin c(6)Spin^c(6) is the connected double cover of U(4)U(4) corresponding to the (unique) index-2 subgroup 2π 1(U(4))2\mathbb{Z} \hookrightarrow \mathbb{Z} \simeq \pi_1(U(4)). This is because the multiplication map is surjective and has kernel canonically isomorphic to μ 4\mu_4, the fourth roots of unity, via the map μ 4SU(4)×U(1)\mu_4\to SU(4)\times U(1) sending ζ(ζ 1I,ζ)\zeta\mapsto (\zeta^{-1}I,\zeta). Hence there are a pair of 2:1 surjective homomorphisms

    SU(4)×U(1)Spin c(6)U(4), SU(4) \times U(1) \to Spin^c(6) \to U(4),

    and hence the result.

The last two examples can also be considered as exceptional isomorphisms, even if not to one of the more classical Lie groups.


Group extension


We have a short exact sequence

U(1)Spin cSO, U(1) \to Spin^c \to SO \,,

where U(1)Spin cU(1) \to Spin^c is the canonical inclusion into the defining product U(1)Spin×U(1)Spin× 2U(1)U(1) \to Spin \times U(1) \to Spin \times_{\mathbb{Z}_2} U(1).


As the homotopy fiber of the smooth W 3\mathbf{W}_3

We discuss in the following that

  1. the universal third integral Stiefel-Whitney class W 3W_3 has an essentially unique lift from ∞Grpd \simeq Top to Smooth∞Grpd;

  2. the smooth delooping BSpin cSmoothGrpd\mathbf{B}Spin^c \in Smooth\infty Grpd is the homotopy fiber of W 3\mathbf{W}_3, hence is the circle 2-bundle over BSO\mathbf{B} SO classified by W 3\mathbf{W}_3.


We have a homotopy pullback diagram

BSpin c Bdet BU(1) c 1mod2 BSO w 2 B 2 2 \array{ \mathbf{B} Spin^c &\stackrel{\mathbf{B}det}{\longrightarrow}& \mathbf{B}U(1) \\ \big\downarrow && \big\downarrow{}^{\mathrlap{\mathbf{c}_1 mod 2}} \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\longrightarrow}& \mathbf{B}^2 \mathbb{Z}_2 }

in Smooth∞Grpd, where


We present the sitation as usual in the projective model structure on simplicial presheaves over CartSp by ∞-anafunctors.

The first Chern class is given by the ∞-anafunctor

B() c 1 B(1)=B 2 BU(1), \array{ \mathbf{B}(\mathbb{Z} \to \mathbb{R}) &\stackrel{\mathbf{c}_1}{\to}& \mathbf{B}(\mathbb{Z} \to 1) = \mathbf{B}^2 \mathbb{Z} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B} U(1) } \,,

where (G 1G 0)(G_1 \to G_0) denotes a presentation of a strict 2-group by a crossed module.

The second Stiefel-Whitney class is given by

B( 2Spin) w 2 B( 21)=B 2 2 BSO. \array{ \mathbf{B}(\mathbb{Z}_2 \to Spin) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}(\mathbb{Z}_2 \to 1) = \mathbf{B}^2 \mathbb{Z}_2 \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B} SO } \,.

Notice that the top horizontal morphism here is a fibration.

Therefore the homotopy pullback in question is (as discussed there) given by the ordinary pullback QQ in

Q B() B( 2Spin) B 2 2. \array{ Q &\to& \mathbf{B}(\mathbb{Z} \to \mathbb{R}) \\ \downarrow && \downarrow \\ \mathbf{B}(\mathbb{Z}_2 \to Spin) &\to& \mathbf{B}^2 \mathbb{Z}_2 } \,.

This pullback is B(Spin×)\mathbf{B}(\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R}), where

:n(nmod2,n). \partial\colon n \mapsto ( n \,mod\, 2 , n) \,.

This is equivalent to

(Spin×) ( 2Spin×(/2)) ( 2Spin×U(1)), \begin{aligned} (\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R}) & \simeq (\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times (\mathbb{R}/2\mathbb{Z})) \\ & \simeq (\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1)) \end{aligned} \,,

(notice the non-standard identification U(1)/(2)U(1) \simeq \mathbb{R}/(2\mathbb{Z}) here, which is important below in prop. for the identification of detdet) where now \partial' is the diagonal embedding of the subgroup

:σ(σ,σ). \partial'\colon \sigma \mapsto (\sigma, \sigma) \,.

This in turn is equivalent to

B(Spin× 2U(1)), \mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1)) \,,

which is def. .


Compare this with the similar but different homotopy pullback that defines the spin group

BSpin * BSO w 2 B 2 2 \array{ \mathbf{B}Spin &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 }

Under the identificaton Spin cSpin× 2U(1)Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times} U(1) the “universal determinant line bundle map”

det:Spin cU(1) det \colon Spin^c \to U(1)

is given in components by

(g,c)2c (g,c) \mapsto 2 c

(where on the right we write the group structure additively).


By the proof of prop. the U(1)U(1)-factor in Spin cSpin× 2U(1)Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times}U(1) arises from the identification U(1)/2U(1) \simeq \mathbb{R}/2\mathbb{Z}. But under the horizontal map as it appears in the homotopy pullback in that proof this corresponds to multiplication by 2.


The third integral Stiefel-Whitney class

W 3β 2w 2:BSOw 2B 2 2β 2B 3 W_3 \coloneqq \beta_2 \circ w_2 \colon B SO \stackrel{w_2}{\to} B^2 \mathbb{Z}_2 \stackrel{\beta_2}{\to} B^3 \mathbb{Z}

has an essentially unique lift through geometric realization ||:{\vert-\vert}\colon Smooth∞Grpd Π\stackrel{\Pi}{\to} ∞Grpd \stackrel{\simeq}{\to} Top

given by

W 3=β 2w 2:BSO(n)w 2B 2 2β 2B 2U(1), \mathbf{W}_3 = \mathbf{\beta}_2 \circ \mathbf{w}_2 \colon \mathbf{B} SO(n) \stackrel{w_2}{\to} \mathbf{B}^2 \mathbb{Z}_2 \stackrel{\mathbf{\beta}_2}{\to} \mathbf{B}^2 U(1) \,,

where β 2\mathbf{\beta}_2 is simply given by the canonical subgroup embedding.


Once we establish that this is a lift at all, the essential uniqueness follows from the respective theorem at smooth ∞-groupoid – structures.

The ordinary Bockstein homomorphism β 2\beta_2 is presented by the ∞-anafunctor

B 2(2) B 2(1)=B 3 B 2 2. \array{ \mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z}) &\to& \mathbf{B}^2 (\mathbb{Z} \to 1) = \mathbf{B}^3 \mathbb{Z} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^2 \mathbb{Z}_2 } \,.

Accordingly we need to lift the canonical presentation of β 2\mathbf{\beta}_2 to a comparable \infty-anafunctor. This is accomplished by

B 2(2) β^ 2 B 2(2) B 2 2 β 2 B 2U(1). \array{ \mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z}) &\stackrel{\hat \mathbf{\beta}_2}{\to}& \mathbf{B}^2 (\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{R}) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^2 \mathbb{Z}_2 &\stackrel{\mathbf{\beta}_2}{\to}& \mathbf{B}^2 U(1) } \,.

Here the top horizontal morphism is induced from the morphism of crossed modules that is given by the commuting diagram

id 2 2 . \array{ \mathbb{Z} &\stackrel{id}{\to}& \mathbb{Z} \\ \downarrow^{\mathrlap{\cdot 2}} && \downarrow^{\mathrlap{\cdot 2}} \\ \mathbb{Z} &\stackrel{}{\hookrightarrow}& \mathbb{R} } \,.

Since \mathbb{R} is contractible, we have indeed under geometric realization an equivalence

|B 2(2)| |β^ 2| |B 2(2)| |B 2(2)| |B 2(1)| |B 2 2| β 2 |B 3|. \array{ \vert\mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})\vert &\stackrel{\vert \hat {\mathbf{\beta}}_2\vert}{\to}& \vert \mathbf{B}^2 (\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{R}) \vert \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \vert\mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})\vert &\to& \vert\mathbf{B}^2(\mathbb{Z} \to 1)\vert \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \vert B^2 \mathbb{Z}_2\vert & \stackrel{\beta_2}{\to}& \vert B^3 \mathbb{Z}\vert } \,.

The sequence

BU(1)c 1mod2B 2 2β 2B 2U(1), \mathbf{B} U(1) \stackrel{\mathbf{c}_1 mod 2}{\to} \mathbf{B}^2 \mathbb{Z}_2 \stackrel{\mathbf{\beta}_2}{\to} \mathbf{B}^2 U(1) \,,

where β 2\mathbf{\beta}_2 is the smoothly refined Bockstein homomorphism from prop. , is a fiber sequence.


The homotopy fiber of B 2BU(1)\mathbf{B} \mathbb{Z}_2 \to \mathbf{B}U(1) is U(1)/ 2U(1)U(1)/\mathbb{Z}_2 \simeq U(1). Thinking of this is (1/2)(\mathbb{Z} \stackrel{\cdot 1/2}{\to} \mathbb{R}) one sees that the inclusion of this fiber is indeed c 1mod2\mathbf{c}_1 mod 2.


The delooping BSpin c\mathbf{B}Spin^c of the Lie group Spin cSpin^c in Smooth∞Grpd is the homotopy fiber of the universal third smooth integral Stiefel-Whitney class from .

BSpin cBSOW 3B 2U(1), \mathbf{B}Spin^c \to \mathbf{B} SO \stackrel{\mathbf{W}_3}{\to} \mathbf{B}^2 U(1) \,,

Then consider the pasting diagram of homotopy pullbacks

BSpin c BU(1) * c 1mod2 BSO w 2 B 2 2 β 2 B 2U(1). \array{ \mathbf{B}Spin^c &\to& \mathbf{B} U(1) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} && \downarrow \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 &\stackrel{\mathbf{\beta}_2}{\to}& \mathbf{B}^2 U(1) } \,.

The right square is a homotopy pullback by prop. . The left square is a homotopy pullback by prop. . The bottom composite is the smooth W 3\mathbf{W}_3 by prop .

This implies by claim by the pasting law.

Relation to metaplectic group Mp cMp^c

There is a direct analogy between Spin, Spin^c and the metaplectic groups Mp and Mp^c (see there for more).


For more see the references at spin^c structure.

The exceptional isomorphisms in low dimension are described in

  • Ozbagci, B., Stipsicz, A.I. (2004). Spin cSpin^c Structures on 3- and 4-Manifolds. In: Surgery on Contact 3-Manifolds and Stein Surfaces. Bolyai Society Mathematical Studies, vol 13. Springer, Berlin, Heidelberg. doi:10.1007/978-3-662-10167-4_6.

Last revised on September 17, 2023 at 11:16:57. See the history of this page for a list of all contributions to it.