# Moscow Mathematical Journal

Volume 16, Issue 2, April–June 2016 pp. 237–273.

Topology and Geometry of the Canonical Action of *T*^{4} on the complex Grassmannian *G*_{4,2} and the complex projective space ℂ*P*^{5}

**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: **

We consider the canonical action of the compact torus *T*^{4}
on the complex Grassmann manifold *G*_{4,2} and prove that the orbit space
*G*_{4,2}/*T*^{4} is homeomorphic to the sphere *S*^{5}. We prove that the induced
map from *G*_{4,2} to the sphere *S*^{5} is not smooth and describe its smooth
and singular points. We also consider the action of *T*^{4} on ℂ*P*^{5} induced
by the composition of the second symmetric power representation of *T*^{4} in *T*^{6} and the standard action of *T*^{6} on ℂ*P*^{5} and prove that the
orbit space ℂ*P*^{5}/*T*^{4} is homeomorphic to the join ℂ*P*^{2}∗*S*^{2}. The Plücker
embedding *G*_{4,2} ⊂ ℂ*P*^{5} is equivariant for these actions and induces the
embedding ℂ*P*^{1} ∗*S*^{2} ⊂ ℂ*P*^{2}∗*S*^{2} for the standard embedding ℂ*P*^{1} ⊂ ℂ*P*^{2}.
All our constructions are compatible with the involution given by
the complex conjugation and give the corresponding results for the real
Grassmannian *G*_{4,2}(ℝ) and the real projective space ℝ*P*^{5} for the action
of the group ℤ_{2}^{4}. We prove that the orbit space *G*_{4,2}(ℝ)/ℤ_{2}^{4} is homeomorphic to the sphere *S*^{4} and that the orbit space ℝ*P*^{5}/ℤ_{2}^{4} is homeomorphic
to the join ℝ*P*^{2}∗*S*^{2}.

2010 Math. Subj. Class. 57S25, 57N65, 53D20, 53B20, 14M25, 52B11.

**Keywords:**Torus action, orbit, space, Grassmann manifold, complex projective space.

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