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spin^c

Definition

Definition

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

$$\begin{array}{rl}{\mathrm{Spin}}^c& ≔\mathrm{Spin}{\times }_{{\mathbb{Z}}_2}U(1)\\ & =(\mathrm{Spin}\times U(1))/{\mathbb{Z}}_2,\end{array}$$\begin{aligned} Spin^c & \coloneqq Spin \times_{\mathbb{Z}_2} U(1) \\ & = (Spin \times U(1))/{\mathbb{Z}_2} \,, \end{aligned}

of the product of the spin group with the circle group by the common subgroup of order 2 ${\mathbb{Z}}_2\hookrightarrow \mathbb{Z}$ and ${\mathbb{Z}}_2\hookrightarrow U(1)$.

Properties

Proposition

We have a short exact sequence

$$U(1)\to {\mathrm{Spin}}^c\to \mathrm{SO},$$U(1) \to Spin^c \to SO \,,

where $U(1)\to {\mathrm{Spin}}^c$ is the canonical inclusion into the defining product $U(1)\to \mathrm{Spin}\times U(1)\to \mathrm{Spin}{\times }_{{\mathbb{Z}}_2}U(1)$.

General

As the homotopy fiber of the smooth $w_2$

We dicuss in the following that

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

• the smooth delooping $B{\mathrm{Spin}}^c\in \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}$ is the homotopy fiber of $W_3$, hence is the circle 2-bundle over $B\mathrm{SO}$ classified by $W_3$.

Proposition

We have a homotopy pullback diagram

$$\begin{array}{ccc}B{\mathrm{Spin}}^c& \to & BU(1)\\ \downarrow & & {\downarrow }^{c_1\mathrm{mod}2}\\ B\mathrm{SO}& \stackrel{w_2}{\to }& B^2{\mathbb{Z}}_2\end{array}$$\array{ \mathbf{B} Spin^c &\to& \mathbf{B}U(1) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 }

in Smooth∞Grpd, where

• the right morphism is the universal first Chern class modulo 2;

• the bottom morphism is the universal second Stiefel-Whitney class.

Proof

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

$$\begin{array}{ccc}B(\mathbb{Z}\to ℝ)& \stackrel{c_1}{\to }& B(\mathbb{Z}\to 1)=B^2\mathbb{Z}\\ {\downarrow }^{\simeq }\\ BU(1)\end{array},$$\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 $(G_1\to G_0)$ denotes a presentation of a strict 2-group by a crossed module.

The second Stiefel-Whitney class is given by

$$\begin{array}{ccc}B({\mathbb{Z}}_2\to \mathrm{Spin})& \stackrel{w_2}{\to }& B({\mathbb{Z}}_2\to 1)=B^2{\mathbb{Z}}_2\\ {\downarrow }^{\simeq }\\ B\mathrm{SO}\end{array}.$$\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 $Q$ in

$$\begin{array}{ccc}Q& \to & B(\mathbb{Z}\to ℝ)\\ \downarrow & & \downarrow \\ B({\mathbb{Z}}_2\to \mathrm{Spin})& \to & B^2{\mathbb{Z}}_2\end{array}.$$\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 $B(\mathbb{Z}\stackrel{\partial }{\to }\mathrm{Spin}\times ℝ)$, where

$$\partial :n\mapsto (n\mathrm{mod}2,n).$$\partial\colon n \mapsto ( n mod 2 , n) \,.

This is equivalent to

$$B({\mathbb{Z}}_2\stackrel{\partial \prime }{\to }\mathrm{Spin}\times U(1))$$\mathbf{B}(\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1))

where now $\partial \prime$ is the diagonal embedding of the subgroup

$$\partial \prime :\sigma \mapsto (\sigma ,\sigma ).$$\partial'\colon \sigma \mapsto (\sigma, \sigma) \,.

This in turn is equivalent to

$$B(\mathrm{Spin}{\times }_{{\mathbb{Z}}_2}U(1)),$$\mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1)) \,,

which is def. 1.

Remark

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

$$\begin{array}{ccc}B\mathrm{Spin}& \to & *\\ \downarrow & & \downarrow \\ B\mathrm{SO}& \stackrel{w_2}{\to }& B^2{\mathbb{Z}}_2\end{array}$$\array{ \mathbf{B}Spin &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 }

Proposition

The third integral Stiefel-Whitney class

$$W_3≔{\beta }_2\circ w_2:B\mathrm{SO}\stackrel{w_2}{\to }{B}^2\mathbb{Z}\stackrel{{\beta }_2}{\to }{B}^3\mathbb{Z}$$W_3 \coloneqq \beta_2 \circ w_2 \colon B SO \stackrel{w_2}{\to} B^2 \mathbb{Z} \stackrel{\beta_2}{\to} B^3 \mathbb{Z}

has an essentially unique lift through geometric realization $\mid -\mid :$ Smooth∞Grpd $\stackrel{\Pi }{\to }$ ∞Grpd $\stackrel{\simeq }{\to }$ Top

given by

$$W_3={\beta }_2\circ w_2:B\mathrm{SO}(n)\stackrel{w_2}{\to }{B}^2{\mathbb{Z}}_2\stackrel{{\beta }_2}{\to }{B}^{2}U(1),$$\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 ${\beta }_2$ is simply given by the canonical subgroup embedding.

Proof

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 ${\beta }_2$ is presented by the ∞-anafunctor

$$\begin{array}{ccc}{B}^2(\mathbb{Z}\stackrel{\cdot 2}{\to }\mathbb{Z})& \to & B^2(\mathbb{Z}\to 1)=B^3\mathbb{Z}\\ {\downarrow }^{\simeq }\\ B^2{\mathbb{Z}}_2\end{array}.$$\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 ${\beta }_2$ to a comparable $\mathrm{\infty }$-anafunctor. This is accomplished by

$$\begin{array}{ccc}{B}^2(\mathbb{Z}\stackrel{\cdot 2}{\to }\mathbb{Z})& \stackrel{{\hat{\beta }}_2}{\to }& B^2(\mathbb{Z}\stackrel{\cdot 2}{\to }ℝ)\\ {\downarrow }^{\simeq }& & {\downarrow }^{\simeq }\\ B^2{\mathbb{Z}}_2& \stackrel{{\beta }_2}{\to }& B^{2}U(1)\end{array}.$$\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

$$\begin{array}{ccc}\mathbb{Z}& \stackrel{\mathrm{id}}{\to }& \mathbb{Z}\\ {\downarrow }^{\cdot 2}& & {\downarrow }^{\cdot 2}\\ \mathbb{Z}& \stackrel{}{\hookrightarrow }& ℝ\end{array}.$$\array{ \mathbb{Z} &\stackrel{id}{\to}& \mathbb{Z} \\ \downarrow^{\mathrlap{\cdot 2}} && \downarrow^{\mathrlap{\cdot 2}} \\ \mathbb{Z} &\stackrel{}{\hookrightarrow}& \mathbb{R} } \,.

Since $ℝ$ is contractible, we have indeed under geometric realization an equivalence

$$\begin{array}{ccc}\mid B^2(\mathbb{Z}\stackrel{\cdot 2}{\to }\mathbb{Z})\mid & \stackrel{\mid {\hat{\beta }}_2\mid }{\to }& \mid B^2(\mathbb{Z}\stackrel{\cdot 2}{\to }ℝ)\mid \\ {\downarrow }^{\simeq }& & {\downarrow }^{\simeq }\\ \mid B^2(\mathbb{Z}\stackrel{\cdot 2}{\to }\mathbb{Z})\mid & \to & \mid B^2(\mathbb{Z}\to 1)\mid \\ {\downarrow }^{\simeq }& & {\downarrow }^{\simeq }\\ \mid B^2{\mathbb{Z}}_2\mid & \stackrel{{\beta }_2}{\to }& \mid B^3\mathbb{Z}\mid \end{array}.$$\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 } \,.

Proposition

The sequence

$$BU(1)\stackrel{c_1\mathrm{mod}2}{\to }{B}^2{\mathbb{Z}}_2\stackrel{{\beta }_2}{\to }{B}^{2}U(1),$$\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 ${\beta }_2$ is the smoothly refined Bockstein homomorphism from prop. 3, is a fiber sequence.

Proof

The homotopy fiber of $B{\mathbb{Z}}_2\to BU(1)$ is $U(1)/{\mathbb{Z}}_2\simeq U(1)$. Thinking of this is $(\mathbb{Z}\stackrel{\cdot 1/2}{\to }ℝ)$ one sees that the inclusion of this fiber is indeed $c_1\mathrm{mod}2$.

Proposition

The delooping $B{\mathrm{Spin}}^c$ of the Lie group ${\mathrm{Spin}}^c$ in Smooth∞Grpd is the homotopy fiber of the universal third smooth integral Stiefel-Whitney class from 3.

$$B{\mathrm{Spin}}^c\to B\mathrm{SO}\stackrel{W_3}{\to }{B}^{2}U(1),$$\mathbf{B}Spin^c \to \mathbf{B} SO \stackrel{\mathbf{W}_3}{\to} \mathbf{B}^2 U(1) \,,

Proof

Then consider the pasting diagram of homotopy pullbacks

$$\begin{array}{ccccc}B{\mathrm{Spin}}^c& \to & BU(1)& \to & *\\ \downarrow & & {\downarrow }^{c_1\mathrm{mod}2}& & \downarrow \\ B\mathrm{SO}& \stackrel{w_2}{\to }& B^2{\mathbb{Z}}_2& \stackrel{{\beta }_2}{\to }& B^{2}U(1)\end{array}.$$\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. 4. The left square is a homotopy pullback by prop. 2. The bottom composite is the smooth $W_3$ by prop 3.

This implies by claim by the pasting law.

Related concepts

• spin group, ${\mathrm{Spin}}^c$-group, spin structure

• spin^c structure, twisted spin^c structure

• string 2-group, string structure

• string^c 2-group