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## Tsuchiya : On homology 3-spheres defined by two knots

The book gives a systematic exposition of diverse ideas and methods in the area, from algebraic topology of manifolds to invariants arising from quantum field theories. The main topics covered are constructions and classification of homology 3-spheres, Rokhlin invariant, Casson invariant and its extensions, including invariants of Walker and Lescop, Herald and Lin invariants of knots, and equivariant Casson invariants, Floer homology and gauge-theoretical invariants of homology cobordism. Many of the topics covered in the book appear in monograph form for the first time. The book gives a rather broad overview of ideas and methods and provides a comprehensive bibliography.

## Homology sphere

It will be appealing to both graduate students and researchers in mathematics and theoretical physics. A simple construction of this space begins with a dodecahedron. Each face of the dodecahedron is identified with its opposite face, using the minimal clockwise twist to line up the faces.

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Gluing each pair of opposite faces together using this identification yields a closed 3-manifold. See Seifertâ€”Weber space for a similar construction, using more "twist", that results in a hyperbolic 3-manifold. One can also pass instead to the universal cover of SO 3 which can be realized as the group of unit quaternions and is homeomorphic to the 3-sphere.

Another approach is by Dehn surgery. If A is a homology 3-sphere not homeomorphic to the standard 3-sphere, then the suspension of A is an example of a 4-dimensional homology manifold that is not a topological manifold.

## Invariants of Homology 3-Spheres

The double suspension of A is homeomorphic to the standard 5-sphere, but its triangulation induced by some triangulation of A is not a PL manifold. In other words, this gives an example of a finite simplicial complex that is a topological manifold but not a PL manifold.

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It is not a PL manifold because the link of a point is not always a 4-sphere. As of [update] the existence of such a homology 3-sphere was an unsolved problem.

On March 11, , Ciprian Manolescu posted a preprint on the ArXiv [5] claiming to show that there is no such homology sphere with the given property, and therefore, there are 5-manifolds not homeomorphic to simplicial complexes.