The homotopy class of non-singular Morse-Smale vector fields on 3-manifolds

NMS Flows on S 3 with no Heteroclinic Trajectories Connecting Saddle Orbits

“…In order to classify the set of closed orbits of a NMS flow on S 3 , M. Wada [8] describes the different types of fat round 1-handles for the manifold S 3 . This result, was independently obtained in a different way by K. Yano [9].…”

“…In dimension three, only the set of periodic orbits of non singular Morse-Smale systems (NMS) on S 3 (M. Wada [8], K. Yano [9]) and S 2 × S 1 (A. Cordero, J. Martínez Alfaro and P. Vindel [5]) have been characterized in terms of links. These characterizations are based on the round handle decomposition (RHD) introduced by Asimov [1] and Morgan [6].…”

Round-handle decomposition ofS2×S1

“…In fact, the round-handle decomposition of S 3 has been obtained by Wada [5] and Yano [6,7,17]. Seeing that, the round-handle decomposition of S 2 Â S 1 is interesting for many reasons: the prime decomposition theorem establishes the decomposition of every compact 3-manifold in terms of a connected sum of prime manifolds.…”

On the number of periodic orbits of Morse-Smale flows on graph manifolds

“…For an introduction to basic notions on Morse-Smale vector fields we point the reader to [10, §1]. In [10], Yano determined which homotopy classes of non-singular vector fields of Y admit a non-singular Morse-Smale (in the following just nMS) representative. As a consequence of his work and the work of Wilson from [9], it follows that for a graph manifold Y there exists a finite number n(Y ) such that in every homotopy class of non-singular vector fields there is a Morse-Smale vector field whose number of periodic orbits is less or equal to n(Y ) (see [10,Remark 5.2]).…”

“…In [10], Yano determined which homotopy classes of non-singular vector fields of Y admit a non-singular Morse-Smale (in the following just nMS) representative. As a consequence of his work and the work of Wilson from [9], it follows that for a graph manifold Y there exists a finite number n(Y ) such that in every homotopy class of non-singular vector fields there is a Morse-Smale vector field whose number of periodic orbits is less or equal to n(Y ) (see [10,Remark 5.2]). Furthermore, Yano remarked there that it would be interesting to determine these numbers or to find a relation between a homotopy class h of non-singular vector fields and the number n(Y, h) which is defined as the minimal number of periodic orbits a nMS vector field in the class h admits.…”