Moscow Mathematical Journal
Volume 16, Issue 2, April–June 2016 pp. 237–273.
Topology and Geometry of the Canonical Action of T4 on the complex Grassmannian G4,2 and the complex projective space ℂP5
We consider the canonical action of the compact torus T4
on the complex Grassmann manifold G4,2 and prove that the orbit space
G4,2/T4 is homeomorphic to the sphere S5. We prove that the induced
map from G4,2 to the sphere S5 is not smooth and describe its smooth
and singular points. We also consider the action of T4 on ℂP5 induced
by the composition of the second symmetric power representation of T4 in T6 and the standard action of T6 on ℂP5 and prove that the
orbit space ℂP5/T4 is homeomorphic to the join ℂP2∗S2. The Plücker
embedding G4,2 ⊂ ℂP5 is equivariant for these actions and induces the
embedding ℂP1 ∗S2 ⊂ ℂP2∗S2 for the standard embedding ℂP1 ⊂ ℂP2.
All our constructions are compatible with the involution given by
the complex conjugation and give the corresponding results for the real
Grassmannian G4,2(ℝ) and the real projective space ℝP5 for the action
of the group ℤ24. We prove that the orbit space G4,2(ℝ)/ℤ24 is homeomorphic to the sphere S4 and that the orbit space ℝP5/ℤ24 is homeomorphic
to the join ℝP2∗S2. 2010 Math. Subj. Class. 57S25, 57N65, 53D20, 53B20, 14M25, 52B11.
Authors:
Victor M. Buchstaber (1) and Svjetlana Terzić (2)
Author institution:(1) Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina Street 8, 119991 Moscow, Russia
(2) Faculty of Science, University of Montenegro Dzordza Vasingtona bb, 81000 Podgorica, Montenegro
Summary:
Keywords: Torus action, orbit, space, Grassmann manifold, complex projective space.
Contents
Full-Text PDF