A subset \(W\) of a topological \(n\)-manifold \(M\) is \(S^1\)-contractible if there are continuous maps \(f : W\to S^1\) and \(\alpha : S^1\to M\) such that the inclusion map \(i : W\to M\) is homotopic to \(\alpha \circ f\). The \(S^1\)-category of \(M\), denoted by \(\mathit{cat}_{S^1}M\), is the smallest number of subsets of \(M\) which are open, \(S^1\)-contractible, and needed to cover \(M\). If the \(S^1\)’s above are replaced with \(\mathcal{A}\)’s, then we obtain the notions \(\mathcal{A}\)-contractible and \(\mathcal{A}\)-category in the same way. In particular, if \(\mathcal{A}\) is the one point set \(\{ p\}\), then the \(\{ p\}\)-category is known as the Lyusternik-Shnirel’man category. These concepts were introduced to study decomposition of \(M\) into simple parts. Note that if \(M\) is closed with \(n\geq 2\), then \(2\leq \mathit{cat}_{S^1}M\leq n+1\).
A strong result is known. A closed \(3\)-manifold \(M^3\) has \(\mathit{cat}_{S^1}M^3=2\) if and only if the fundamental group \(\pi_1(M^3)\) is cyclic. The condition was topologically characterized by P. Olum [Ann. Math. (2) 58, 458-480 (1953; Zbl 0052.19901)] and G. Perelman showed that \(\pi_1(M^3)\) is cyclic if and only if \(M^3\) is a lens space including \(S^3\) and \(S^1\times S^2\), or the non-orientable \(S^2\)-bundle over \(S^1\), cf. G. Perelman [The entropy formula for the Ricci flow and its geometric applications. arXiv e-print service, Cornell University Library, Paper No. 0211159, electronic only (2002; Zbl 1130.53001); Ricci flow with surgery on three-manifolds. arXiv e-print service, Cornell University Library, Paper No. 0303109, electronic only (2003; Zbl 1130.53002); Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv e-print service, Cornell University Library, Paper No. 0307245, electronic only (2003; Zbl 1130.53003)] and also [J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture. Clay Mathematics Monographs 3. Providence, RI: American Mathematical Society (AMS); Cambridge, MA: Clay Mathematics Institute. (2007; Zbl 1179.57045)].
The authors consider the case of \(3\)-manifolds with boundary. It follows from Perelman [loc. cit.] that a \(3\)-manifold \(M^3\) with \(\partial M^3\neq \emptyset\) has \(\mathit{cat}_{S^1}M^3=1\) if and only if \(M^3\) is a ball, a solid torus, or a solid Klein bottle. Cyclicity of \(\pi_1(M^3)\) also characterizes the condition. The main theorem (Theorem 1) is : (a)A compact, orientable, irreducible \(3\)-manifold \(M^3\) with \(\mathit{cat}_{S^1}M^3=2\) is a Seifert fiber space with handles and with at most two exceptional fibers. (b)A Seifert fiber space with handles and with at most two exceptional fibers \(M^3\) has \(\mathit{cat}_{S^1}M^3\leq 2\). Here a Seifert fiber space with handles is obtained from a Seifert fiber space by attaching \(1\)-handles to the boundary. A corollary (Corollary 3) states : (a) The exterior \(E(L)\) of a non-splittable link \(L\) with at least two components in \(S^3\) has \(\mathit{cat}_{S^1}E(L)=2\) if and only if \(L\) is a Burde-Murasugi link. (b)The exterior \(E(K)\) of a knot \(K\) in \(S^3\) has \(\mathit{cat}_{S^1}E(K)=2\) if and only if \(K\) is a non-trivial torus knot. Here a Burde-Murasugi link is a link \(L\) in \(S^3\) consisting of fibers of some Seifert fibering of \(S^3\).
For the \(3\)-dimensional case, we can work in the PL-category. Suppose \(\mathit{cat}_{S^1}M^3=2\). Then we can take subsets \(W_0\) and \(W_1\) of \(M^3\) such that \(W_i\)\((i=0, 1)\) is open and \(S^1\)-contractible, and \(M^3=W_0\cup W_1\). Lemma 1, Lemma 2 and Corollary 1 state that we can replace \(W_i\)\((i=0, 1)\) with a PL-submanifold satisfying “good” conditions. Corollary 1 is the crucial one saying that every connected component of \(W_i\) is a ball or a solid torus. These are proved by using effectively the homotopy extension property to construct homotopies. The proof of Theorem 1 is started from the situation in Corollary 1, and completed by elementary topological and homological arguments.