On the Gray index conjecture for phantom maps

We study the Gray index of phantom maps, which is a numerical invariant of phantom maps. It is conjectured that the only phantom map with infinite Gray index between finite-type spaces is the constant map. We disprove this conjecture by constructing a counter example. We also prove that this conjecture is valid if the target spaces of phantom maps are restricted to simply connected finite complexes. As an application of the counter example we show that \SNT^{\infty}(X) can be non-trivial for some space $X$ of finite type.Comment: 18page

Tight complexes are Golod

The Golodness of a simplicial complex is defined algebraically in terms of the Stanley-Reisner ring, and it has been a long-standing problem to find its combinatorial characterization. The tightness of a simplicial complex is a combinatorial analogue of a tight embedding of a manifold into the Euclidean space, and has been studied in connection to minimal manifold triangulations. In this paper, we prove that tight complexes are Golod, and as a corollary, we obtain that for triangulations of closed connected orientable manifolds, the Golodness and the tightness are equivalent.Comment: 17 pages, minor correction