All triangulated d-manifolds satisfy the inequality for d ≥ 3. A triangulated d-manifold is called tight neighborly if it attains equality in this bound. For each d ≥ 3, a (2d+3)-vertex tight neighborly triangulation of the Sd-1-bundle over S1 with β1 = 1 was constructed by Kühnel in 1986. In this paper, it is shown that there does not exist a tight neighborly triangulated manifold with β1 = 2. In other words, there is no tight neighborly triangulation of (Sd-1x S1)#2 or (Sd-1x̲ S1)#2 for d ≥ 3. A short proof of the uniqueness of Kühnel’s complexes for d ≥ 4 under the assumption β1≠ 0 is also presented.
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