The quadratic form \(E_8\) and exotic homology manifolds. (English) Zbl 1109.57016

Mio, Washington; Ranicki, Andrew

Quinn, Frank (ed.) et al., Exotic homology manifolds. Proceedings of the mini-workshop, Oberwolfach, Germany, June 29–July 5, 2003. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 9, 33-66 (2006).

In the early 1990’s Bryant, Ferry, Mio, and Weinberger described an indirect construction of an exotic homology \(n\)-manifold. Such an object while satisfying Poincaré duality is not homotopy equivalent to any closed topological manifold. This paper shows how the construction can be made more explicit by using a specific realization of a \((-1)^n\)-quadratic form representing the surgery problem \(E_{8} \times T^{2n}\).
For the entire collection see [Zbl 1104.57001].

Reviewer: Jonathan Hodgson (Philadelphia)

MSC:

57P99	Generalized manifolds
19J25	Surgery obstructions (\(K\)-theoretic aspects)
57R67	Surgery obstructions, Wall groups
11E81	Algebraic theory of quadratic forms; Witt groups and rings
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)

Keywords:

surgery problem; exotic ENR homology manifold; Poincaré complex