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Given a root system, the Weyl chambers in the co-weight lattice give rise to a real toric variety, called the real toric variety associated with the Weyl chambers. We compute the integral cohomology groups of real toric varieties associated with the Weyl chambers of type Cn and Dn, completing the computation for all classical types.
Every cohomology ring isomorphism between two non-singular complete toric varieties (respectively, two quasitoric manifolds), with second Betti number 2, is realizable by a diffeomorphism (respectively, homeomorphism).
A fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. In a previous paper, the authors provided a new way to find all characteristic maps on a simplicial complex
obtainable by a sequence of wedgings from
.The main idea was that characteristic maps on
theoretically determine all possible characteristic maps on a wedge of
We further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere
vertices, the Picard number Pic
. We call
a seed if
cannot be obtained by wedgings. First, we show that for a fixed positive integer
, there are at most finitely many seeds of Picard number
supporting characteristic maps. As a corollary, the conjecture proposed by V. V. Batyrev in is solved affirmatively.
Secondly, we investigate a systematicmethod to find all characteristic maps on
using combinatorial objects called (realizable) puzzles that only depend on a seed
. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.
We show that three- and four-stage Bott manifolds are classified up to diffeomorphism by their integral cohomology rings. In addition, any cohomology ring isomorphism between two three-stage Bott manifolds can be realized by a diffeomorphism between the Bott manifolds.
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