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Spin(5)

Contents

Context

Group Theory

Spin geometry

Contents

Idea

The spin group in dimension 5.

Properties

Exceptional isomorphism

Proposition

There is an exceptional isomorphism

Spin(5)Sp(2) Spin(5) \;\simeq\; Sp(2)

between Spin(5) and the quaternionic unitary group Sp(2)=U(2,)Sp(2) = U(2,\mathbb{H}).

This is an indirect consequence of triality, see e.g. Čadek-Vanžura 97)

Action on quaternionic Hopf fibration

Proposition

(Spin(5)-equivariance of quaternionic Hopf fibration)

Consider

  1. the Spin(5)-action on the 4-sphere S 4S^4 which is induced by the defining action on 5\mathbb{R}^5 under the identification S 4S( 5)S^4 \simeq S(\mathbb{R}^5);

  2. the Spin(5)-action on the 7-sphere S 7S^7 which is induced under the exceptional isomorphism Spin(5)Sp(2)=U(2,)Spin(5) \simeq Sp(2) = U(2,\mathbb{H}) (from Prop. ) by the canonical left action of U(2,)U(2,\mathbb{H}) on 2\mathbb{H}^2 via S 7S( 2)S^7 \simeq S(\mathbb{H}^2).

Then the quaternionic Hopf fibration S 7h S 4S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4 is equivariant with respect to these actions.

This is almost explicit in Porteous 95, p. 263

Cohomology

Proposition

The integral cohomology ring of the classifying space BSpin(5)B Spin(5) is spanned by two generators

  1. the first fractional Pontryagin class 12p 1\tfrac{1}{2}p_1

  2. the linear combination 12p 212(p 1) 2\tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2 of the half the second Pontryagin class with half the cup product-square of the first Pontryagin class:

H (BSpin(5),)[12p 1,12p 212(p 1) 2] H^\bullet \big( B Spin(5), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \left[ \tfrac{1}{2}p_1, \; \tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2 \right]

(e.g. Kalkkinen 06, Section 3)

Proposition

Let

S 4 BSpin(4) π BSpin(5) \array{ S^4 &\longrightarrow& B Spin(4) \\ && \big\downarrow^{\mathrlap{\pi}} \\ && B Spin(5) }

be the spherical fibration of classifying spaces induced from the canonical inclusion of Spin(4) into Spin(5) and using that the 4-sphere is equivalently the coset space S 4Spin(5)/Spin(4)S^4 \simeq Spin(5)/Spin(4) (this Prop.).

Then the fiber integration of the odd cup powers χ 2k+1\chi^{2k+1} of the Euler class χH 4(BSpin(4),)\chi \in H^4\big( B Spin(4), \mathbb{Z}\big) (see this Prop) are proportional to cup powers of the second Pontryagin class

π *(χ 2k+1)=2(p 2) kH 4(BSpin(5),), \pi_\ast \left( \chi^{2k+1} \right) \;=\; 2 \big( p_2 \big)^k \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,,

for instance

π *(χ) =2 π *(χ 3) =2p 2 π *(χ 5) =2(p 2) 2H 4(BSpin(5),); \begin{aligned} \pi_\ast \big( \chi \big) & = 2 \\ \pi_\ast \left( \chi^3 \right) & = 2 p_2 \\ \pi_\ast \left( \chi^5 \right) & = 2 (p_2)^2 \end{aligned} \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,;

while the fiber integration of the even cup powers χ 2k\chi^{2k} vanishes

π *(χ 2k)=0H 4(BSpin(5),). \pi_\ast \left( \chi^{2k} \right) \;=\; 0 \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,.

(Bott-Cattaneo 98, Lemma 2.1)


rotation groups in low dimensions:

sp. orth. groupspin grouppin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
Spin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)

see also

linebreak

References

Last revised on March 24, 2019 at 10:56:45. See the history of this page for a list of all contributions to it.