Celebratio Mathematica

J. S. Birman and H. M. Hilden: “On the mapping class groups of closed surfaces as covering spaces,” pp. 81–115 in Advances in the theory of Riemann surfaces (Stony Brook, NY, 1969). Edited by L. V. Ahlfors, L. Bers, H. M. Farkas, R. C. Gunning, I. Kra, and H. E. Rauch. Annals of Mathematics Studies 66. Princeton University Press, 1971. MR 0292082 Zbl 0217.48602 incollection

Irwin Kra	Related
Hershel Mark Farkas	Related
Harry Ernest Rauch	Related
Lipman Bers	Related
Lars Valerian Ahlfors	Related
Robert Clifford Gunning	Related
Hugh Michael Hilden	Related

J. S. Birman and H. M. Hilden: “Isotopies of homeomorphisms of Riemann surfaces and a theorem about Artin’s braid group,” Bull. Am. Math. Soc. 78 : 6 (November 1972), pp. 1002–1004. MR 0307217 Zbl 0255.57002 article

J. S. Birman and H. M. Hilden: “Lifting and projecting homeomorphisms,” Arch. Math. (Basel) 23 : 1 (1972), pp. 428–434. MR 0321071 Zbl 0247.55001 article

Let \( X \) be a pathwise connected and locally pathwise connected topological space, \( G \) the group of all self-homeomorphisms of \( X \), and \( D \) the subgroup of maps isotopic to the identity. The homeotopy group of \( X \) is defined to be the group \( G/D \). Let \( \tilde{X} \) be a p.c., l.p.c. covering space of \( X \), with projection \( p \). The relationship between the homeotopy groups of \( \tilde{X} \) and \( X \) is studied. It is shown that under sufficiently strong restrictions on \( \tilde{X} \), \( X \) and \( p \) the homeotopy group of \( X \) is isomorphic to a factor group of the homeotopy group of \( \tilde{X} \), with weaker results as one weakens the restrictions on \( \tilde{X} \) and \( X \).

The situation studied here first came to the authors’ attention in an earlier investigation [Birman and Hilden 1971]. The homeotopy groups of 2-manifolds play an important role in the theory of Riemann surfaces, and also in the classification of 3-manifolds. It was shown in [Birman and Hilden 1971] that one could gain considerable insight into the structure of the homeotopy groups of surfaces by utilizing the fact that any closed compact orientable surface of genus \( g \) with \( (2g + 2) \) points removed can be regarded as a 2-sheeted covering of a \( (2g + 2) \)-punctured sphere, and making use of the known properties of the homeotopy group of the punctured sphere. The development of this relationship suggested that other coverings of more general spaces might also be of interest, thus motivating the present investigation. At the conclusion of this paper a new application to surface topology is discussed briefly. A detailed workingout of this application will be found in [Birman and Chillingworth 1969], which should appear concurrently with the present work.

J. S. Birman and H. M. Hilden: “The homeomorphism problem for \( S^3 \),” Bull. Am. Math. Soc. 79 : 5 (September 1973), pp. 1006–1010. MR 0319180 Zbl 0272.57001 article

Let \( M \) be a closed, orientable 3-manifold which is defined by a Heegaard splitting of genus \( g \). Each such Heegaard splitting may be associated with a self-homeomorphism of a closed, orientable surface of genus \( g \) (the surface homeomorphism is used to define a pasting map) and it will be assumed that this surface homeomorphism is given as a product of standard twist maps [Lickorish 1962] on the surface. We assert:

If \( M \) is defined by a Heegaard splitting of genus \( \leq 2 \), then an effective algorithm exists to decide whether \( M \) is topologically equivalent to the 3-sphere \( S^3 \). This algorithm also applies to a proper subset of all Heegaard splittings of genus \( > 2 \).

This result is of interest because it had not been known whether such an algorithm was possible for \( g\geq 2 \), and also because the algorithm has a possible application in testing candidates for a counterexample to the Poincaré conjecture.

In this note we will describe the algorithm, and sketch a brief proof. Related results about the connections between representations of 3-manifolds as Heegaard splittings, and as branched coverings of \( S^3 \), are summarized at the end of this paper. A detailed report will appear in another journal.

J. S. Birman and H. M. Hilden: “On isotopies of homeomorphisms of Riemann surfaces,” Ann. Math. (2) 97 : 3 (May 1973), pp. 424–439. MR 0325959 Zbl 0237.57001 article

J. S. Birman and H. M. Hilden: “Heegaard splittings of branched coverings of \( S^3 \),” Trans. Am. Math. Soc. 213 (November 1975), pp. 315–352. MR 0380765 Zbl 0312.55004 article

This paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other hand as a branched covering of the 3-sphere. The ability to pass back and forth between these two representations will be applied in several different ways.

It will be established that there is an effective algorithm to decide whether a 3-manifold of Heegard genus 2 is a 3-sphere.
We will show that the natural map from 6-plat representations of knots and links to genus 2 closed oriented 3-manifolds is injective and surjective. This relates the question of whether or not Heegaard splittings of closed, oriented 3-manifolds are “unique” to the question of whether plat representations of knots and links are “unique”.
We will give a counterexample to a conjecture (unpublished) of W. Haken, which would have implied that \( S^3 \) could be identified (in the class of all simply-connected 3-manifolds) by the property that certain canonical presentations for \( \pi_1S^3 \) are always “nice”.

The final section of the paper studies a special class of genus 2 Heegard splittings: the 2-fold covers of \( S^3 \) which are branched over closed 3-braids. It is established that no counterexamples to the “genus 2 Poincaré conjecture” occur in this class of 3-manifolds.

J. S. Birman and H. M. Hilden: “Erratum to ‘Isotopies of homeomorphisms of Riemann surfaces’,” Ann. of Math. 185 : 1 (2017), pp. 345. MR 3583359 Zbl 06686591 article

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