In the present paper, we characterize Fano Bott manifolds up to diffeomorphism in terms of three operations on matrix. More precisely, we prove that given two Fano Bott manifolds X and X ′ , the following conditions are equivalent:(1) the upper triangular matrix associated to X can be transformed into that of X ′ by those three operations; (2) X and X ′ are diffeomorphic;(3) the integral cohomology rings of X and X ′ are isomorphic as graded rings. As a consequence, we affirmatively answer the cohomological rigidity problem for Fano Bott manifolds.
Let n and k be positive integers with n > k. Given a permutation (π 1 , . . . , πn) of integers 1, . . . , n, we consider k-consecutive sums of π, i.e., s i := k−1 j=0 π i+j for i = 1, . . . , n, where we let π n+j = π j . What we want to do in this paper is to know the exact value of msum(n, k) := min max{s i : i = 1, . . . , n} − k(n + 1) 2 : π ∈ Sn ,where Sn denotes the set of all permutations of 1, . . . , n. In this paper, we determine the exact values of msum(n, k) for some particular cases of n and k. As a corollary of the results, we obtain msum(n, 3), msum(n, 4) and msum(n, 6) for any n.
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