Abstract
This chapter describes the isometries of inner product spaces and their geometric applications. Two basic problems in the geometry of manifolds have led to algebraic obstructions groups based on isometries of inner product spaces over the rational integers. The chapter presents an open book decomposition theorem that states that if M2n+1 is a simply connected odd dimensional; manifold then M has an open book decomposition. A closed simply connected manifold of even dimension greater than six has open book decomposition if and only if its signature is zero. In the special case of an open book decomposition of the standard sphere, Sn, with binding a homotopy sphere, the open book is called a fibered knot.
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