Three-dimensional manifolds, skew-Gorenstein rings and their cohomology. (English) Zbl 1237.16007
Summary: Graded skew-commutative rings occur often in practice. Here are two examples: 1) The cohomology ring of a compact three-dimensional manifold. 2) The cohomology ring of the complement of a hyperplane arrangement (the Orlik-Solomon algebra). We present some applications of the homological theory of these graded skew-commutative rings. In particular we find compact oriented 3-manifolds without boundary for which the Hilbert series of the Yoneda Ext-algebra of the cohomology ring of the fundamental group is an explicit transcendental function. This is only possible for large first Betti numbers of the 3-manifold (bigger than – or maybe equal to – 11). We give also examples of 3-manifolds where the Ext-algebra of the cohomology ring of the fundamental group is not finitely generated.
MSC:
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
| 52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |
| 13D40 | Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series |
| 55P62 | Rational homotopy theory |
| 57N65 | Algebraic topology of manifolds |
Keywords:
three-dimensional manifolds; fundamental groups; lower central series; Gorenstein rings; hyperplane arrangements; homotopy Lie algebras; Yoneda Ext-algebras; local rings; cohomology rings; graded skew-commutative ringsReferences:
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