×

Non-singular Smale flows on three-dimensional manifolds and Whitehead torsion. (English) Zbl 1230.37038

A Smale flow is a structurally stable flow with a one-dimensional invariant set. The main result of this paper gives a relationship between the invariant set of a non-singular Smale flow on a compact connected oriented \(3\)-manifold and the Whitehead torsion of the underlying manifold. The main result can be viewed as a generalization of a result of J. M. Franks [“Knots, links, and symbolic dynamics”, Ann. Math. (2) 113, 529–552 (1981; Zbl 0469.58013)].
Section 2 of the paper reviews the definition of Whitehead torsion \(\tau^\rho(Y,Z)\) for a CW complex pair \((Y,Z)\) in a connected CW complex \(X\). It also discusses what it means for two CW complex pairs to be RSC equivalent (relative simple homotopy type on the complement). Two CW complex pairs that are RSC equivalent have the same relative homotopy type.
Section 3 discusses the Conley index of an isolated invariant set for a smooth flow \(\phi_t\) on a smooth compact manifold \(M\). The author uses an early definition of the Conley index based on the notion of an isolating block.
Definition 3.1. A submanifold \(B\) with corners of \(M\) is a an isolating block for the flow \(\phi_t\) if the following hold.
(1) \(\partial B = \partial_+B \cup \partial_0 B \cup \partial_- B\) is a union of submanifolds with boundaries of codimension one.
(2) The vector field \(d\phi_t/dt\) points into \(B\) on \(\partial_+B\) and points out on \(\partial_-B\).
(3) For any \(x \in \partial_0B\), there is an interval \([T_1,T_2]\) containing \(0\) such that \(\phi_{[T_1,T_2]}(x) \subseteq \partial_0 B\), \(\phi_{T_1}(x) \in \partial_+B\) and \(\phi_{T_2}(x) \in \partial_-B\).
The author proves the following.
Theorem 3.2 If two isolating blocks \(B\) and \(B'\) have the same maximal invariant set, then \((B,\partial_-B)\) and \((B',\partial_-B')\) are RSC equivalent.
This allows for the following definition.
Definition 3.3. Let \(B\) be an isolating block of a flow \(\phi_t\) on a manifold \(M\). Then the RSC equivalence class of \((B,\partial_-B)\) is said to be the RSC index pair of its maximal invariant set.
Section 4 discusses hyperbolic chain recurrent sets, and the author constructs RSC index pairs for basic sets of a Smale flow on a \(3\)-manifold. Section 5 states and proves the main result of the paper.
Theorem 5.1. Let \(\phi_t\) be a non-singular Smale flow on an oriented connected compact \(3\)-manifold \(M\). Suppose that \(\rho:\pi_1(M) \rightarrow G\) is a non-trivial homomorphism from the fundamental group of \(M\) to a free abelian group. If \(\tau^\rho(M, \partial_-M) \neq 0\), then \[ \prod_i det(I - B_i^\rho) = \tau^\rho(M,\partial_-M) \prod_j (1 - \rho(\gamma_j)), \] where \(B_i^\rho\) ranges over the linking matrices of all basic sets of index \(1\) with respect to \(\rho\), and \(\gamma_j\) ranges over all the closed orbits which are attractors or repellers.
The proof of the main theorem uses properties of the RSC index pairs proven in Section 4. Corollary 5.2 is an interesting corollary to the main theorem, which can be reduced to a \(3\)-dimensional version of the Poincaré-Bendixson Theorem.
Corollary 5.2. Let \(\phi_t\) be a non-singular Smale flow on an oriented connected compact \(3\)-manifold. If there is a non-trivial homomorphism \(\rho:\pi_1(M) \rightarrow G\) from the fundamental group of \(M\) to a free abelian group \(G\) such that \(\tau^\rho(M,\partial_-M)\) has a non-trivial denominator, then \(\phi_t\) has either an attracting or a repelling orbit.
Section 6 explains the relation with the Alexander polynomial and how the main theorem is a generalization of Franks’ result.

MSC:

37D15 Morse-Smale systems
37C10 Dynamics induced by flows and semiflows
37B30 Index theory for dynamical systems, Morse-Conley indices
37C20 Generic properties, structural stability of dynamical systems
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.

Citations:

Zbl 0469.58013
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1070/RM1986v041n01ABEH003204 · Zbl 0602.57005 · doi:10.1070/RM1986v041n01ABEH003204
[2] Birman, Contemp. Math. 20 pp 1– (1983) · Zbl 0526.58043 · doi:10.1090/conm/020/718132
[3] DOI: 10.1090/S0002-9904-1967-11798-1 · Zbl 0202.55202 · doi:10.1090/S0002-9904-1967-11798-1
[4] Jiang, Topics in Knot Theory pp 211– (1993) · doi:10.1007/978-94-011-1695-4_11
[5] DOI: 10.2307/1971077 · Zbl 0295.57010 · doi:10.2307/1971077
[6] Ghrist, Knots and Links in Three-Dimensional Flows (1997) · Zbl 0869.58044 · doi:10.1007/BFb0093387
[7] Sánchez-Morgado, Ergod. Th. & Dynam. Sys. 16 pp 405– (1996) · Zbl 0872.58041 · doi:10.1017/S0143385700008877
[8] Fried, Contemp. Math. 58 pp 19– (1987) · doi:10.1090/conm/058.3/893856
[9] DOI: 10.1016/0022-0396(73)90018-1 · Zbl 0238.58010 · doi:10.1016/0022-0396(73)90018-1
[10] DOI: 10.1090/S0002-9947-1969-0251747-9 · doi:10.1090/S0002-9947-1969-0251747-9
[11] Franks, Homology and Dynamical Systems (1982) · doi:10.1090/cbms/049
[12] DOI: 10.1090/S0002-9904-1966-11484-2 · Zbl 0147.23104 · doi:10.1090/S0002-9904-1966-11484-2
[13] DOI: 10.2307/2006996 · Zbl 0469.58013 · doi:10.2307/2006996
[14] DOI: 10.2307/1970268 · Zbl 0108.36502 · doi:10.2307/1970268
[15] Conley, Isolated Invariant Set and the Morse Index (1978) · doi:10.1090/cbms/038
[16] DOI: 10.1016/0022-0396(72)90013-7 · Zbl 0242.54041 · doi:10.1016/0022-0396(72)90013-7
[17] DOI: 10.1016/0022-0396(72)90012-5 · Zbl 0242.58005 · doi:10.1016/0022-0396(72)90012-5
[18] DOI: 10.1016/S0166-8641(99)00069-3 · Zbl 0964.37019 · doi:10.1016/S0166-8641(99)00069-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.