Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
This should have an entry of its own (as it does also on Wikipedia) for ease of linking with triality etc.
added pointer to
for discussion of the cohomology of BSpin(8)
I was really looking for answer to the following question, but not sure yet:
What is the restriction of the Euler class on BSpin(8) to BSpin(5), expressed in terms of the generators of H•(BSpin(5),ℤ)?
added statement of the cohomology ring of BSpin(8):
The ordinary cohomology ring of the classifying space BSpin(8) is:
1) with coefficients in the cyclic group of order 2:
H•(BSpin(8),ℤ2)≃ℤ[w4,w6,w7,w8,ρ2(14(p2−(12p1)2−2χ))]where wi are the universal Stiefel-Whitney classes,
and where
ρ2:H•(BSpin(8),ℤ)→H•(BSpin(8),ℤ2)2) with coefficients in the integers:
H•(BSpin(8),ℤ)≃ℤ[12p1,14(p2−(12p1)2−2χ),χ,δw6]/⟨2δw6⟩,where p1 is the first fractional Pontryagin class, p2 is the second Pontryagin class, χ is the Euler class.
Moreover, we have the relations
ρ2(12p1)=w4ρ2(χ)=w8I have added pointer to
and added two of its Propositions to the text here, expanding on those distinct Spin(7)-subgroups of Spin(8) intersecting in G2.
Now I have a question: By that sequence of pulback squares we should get analogous distinct subgroup inclusions (in distinct conjugacy classes) of SU(3) in Spin(6) and of SU(2) in Spin(5)?!
added pointers to
Robert Bryant, Remarks on Spinors in Low Dimension (pdf, BryantRemarksOnSpinorsInLowDimension.pdf:file)
Megan M. Kerr, New examples of homogeneous Einstein metrics, Michigan Math. J. Volume 45, Issue 1 (1998), 115-134 (euclid:1030132086)
also pointer to
I have added (here and here, respectively) the statements that
The three groups
Sp(1)⋅Sp(2),
Sp(2)⋅Sp(1),
SO(3)×SO(5)
with their canonical embeddings into SO(8) represent three distinct conjugacy classes of subgroups of SO(8) and triality permutes these three transitively;
The three groups
Sp(1)⋅Sp(2),
Sp(2)⋅Sp(1),
Spin(3)⋅Spin(5)
with their canonical embeddings into Spin(8) represent three distinct conjugacy classes of subgroups of Spin(8) and triality permutes these three transitively;
Hope I got this right. This is pretty subtle.
added statement of the action of triality on the universal characteristic classes, paraphrasing Čadek-Vanžura 97, Lemma 4.2:
Consider the delooping of the triality automorphism relating Sp(2).Sp(1) with Spin(5).Spin(3), from above, on classifying spaces
B(Spin(5)⋅Spin(3))↪BSpin(8)↓↓BtriB(Sp(2)⋅Sp(1))↪BSpin(8)Then the pullback of the universal characteristic classes of BSpin(8) (from above) along Btri is as follows:
(Btri)*:12p1↦12p1χ↦14p2−(14p1)2+12χ14p2−(14p1)2−12χ↦−(14p2−(14p1)2−12χ)1 to 14 of 14