nLab Spinᶜ(6)

Idea

$Spin^\mathrm{c}(6)$ is the fourth spinᶜ group. It is used to describe spinᶜ structures on orientable 6-manifolds.

Properties

Proposition

One has a canonical double cover of the unitary group in dimension 4:

$$Spin^\mathrm{c}(6) \twoheadrightarrow U(4).$$

Proof

Using the exceptional isomorphism $Spin(6)\cong SU(4)$ yields:

$$Spin^\mathrm{c}(6) \cong(Spin(6)\times U(1))/\mathbb{Z}_2 =(SU(4)\times U(1))/\mathbb{Z}_2 \twoheadrightarrow(SU(4)\times U(1))/\mathbb{Z}_4 \cong U(4).$$

(In general, one has $(SU(n)\times U(1))/\mathbb{Z}_n\cong U(n)$ using the homomorphism theorem? on the group homomorphism $SU(n)\times U(1)\rightarrow U(n),(U,z)\mapsto Uz$, which is surjective and has $\{(\zeta_n^k\mathbf{1}_n,\zeta_n^{-k})|[k]\in\mathbb{Z}_n \}\cong\mathbb{Z}_n$ as kernel.)

A double cover is the same as an principal O(1)-bundle and corresponds uniquely to a real line bundle $Spin^\mathrm{c}(6)\times_{O(1)}\mathbb{R}\twohearightarrow U(4)$ with the balanced product. Both are classified uniquely by the first Stiefel-Whitney class $w_1(Spin^\mathrm{c}(6))\in H^1(U(4),\mathbb{Z}_2)$, as it is a group isomorphism? over CW complexes like $U(4)$. Using the universal coefficient theorem and the Hurewicz theorem yields:

$$H^1(U(4),\mathbb{Z}_2) \cong Hom(H_1(U(4),\mathbb{Z}),\mathbb{Z}_2) \cong Hom(\pi_1^\mathrm{ab}(U(4)),\mathbb{Z}_2) \cong Hom(\mathbb{Z},\mathbb{Z}_2) \cong\mathbb{Z}_2.$$

($\pi_1^\mathrm{ab}(U(4))\cong\mathbb{Z}$ follows from the determinant $det\colon U(4)\rightarrow U(1)$ inducing a group isomorphism? on the fundamental group.) Now there are two possibilities for the above first Stiefel-Whitney class, which are the trivial and the unique non-trivial one. Since the former would lead to the Lie group isomorphism? $Spin^\mathrm{c}(6)\cong U(4)\times O(1)$ between a connected and not connected space, it has to be the latter. Its classifying map can also be stated easily:

$$i\circ det\colon U(4)\twoheadrightarrow U(1)\cong S^1\cong\mathbb{R}P^1\hookrightarrow\mathbb{R}P^\infty\cong BO(1)$$

The first pullback along the canonical inclusion $i\colon S^1\hookrightarrow BO(1)$ is the real Hopf fibration $i^*S^\infty\cong i^*EO(1)\cong S^1\twoheadrightarrow S^1$ with unique non-trivial first Stiefel-Whitney class $w_1(S^1)\in H^1(S^1,\mathbb{Z}_2)\cong\mathbb{Z}_2$. (Or better denoted as that of the tautological line bundle $\gamma_{\mathbb{R}}^{1,1}=S^1\times_{O(1)}\mathbb{R}$, since the identically denoted first Stiefel-Whitney class of the tangent bundle is trivial.) The second pullback along the determinant (which is not the determinant line bundle) preserves this. Now there is an alternative description:

$$Spin^\mathrm{c}(6) \cong\{(U,z)\in U(4)\times U(1)|det(U)=z^2\}.$$

and the double cover can be easily spotted in the missing sign of the complex number.

Related concepts

• U(2) (Spinᶜ(3), Spinʰ(2))

• Spinᶜ(4)