Dehn surgery on Anosov flows. (English) Zbl 0532.58021

Goodman, Sue

Geometric dynamics, Proc. int. Symp., Rio de Janeiro/Brasil 1981, Lect. Notes Math. 1007, 300-307 (1983).

[For the entire collection see Zbl 0511.00026.]
A flow on a manifold M is Anosov if the tangent bundle of M has an invariant splitting as a direct sum of a contracting and an expanding part \((=hyperbolic\) structure). For a long time, only standard examples were known. Along the lines of work by M. Handel and W. Thurston [Invent. Math. 59, 95-103 (1980; Zbl 0435.58019)], the present note gives a construction of new Anosov 3-manifolds from a given one. This is achieved by modifying an Anosov 3-manifold in a neighborhood of a periodic orbit. The modification, Dehn surgery, removes a solid torus and glues it back in a different way, a construction well known in 3-manifold theory, especially knot theory.

Reviewer: D.Erle

MSC:

37D99	Dynamical systems with hyperbolic behavior
37C10	Dynamics induced by flows and semiflows
57M25	Knots and links in the \(3\)-sphere (MSC2010)
57R65	Surgery and handlebodies

Keywords:

non-algebraic Anosov flows; figure eight knot; hyperbolic periodic orbit; Anosov diffeomorphism; Anosov 3-manifolds