Celebratio Mathematica

M. G. Scharlemann and L. C. Siebenmann: “The Hauptvermutung for \( C^{\infty} \) homeomorphisms, II: A proof valid for open 4-manifolds,” Compositio Math. 29 : 3 (1974), pp. 253–264. MR 375338 Zbl 0315.57010 article

M. G. Scharlemann: A fake homotopy structure on \( S^3\times S^1\mathbin{\#} S^1\times S^2 \) and a structure theorem for five-manifolds. Ph.D. thesis, University of California, Berkeley, 1974. Advised by R. Kirby. MR 2940416 phdthesis

R. C. Kirby and M. G. Scharlemann: “A curious category which equals TOP,” pp. 93–97 in Manifolds—Tokyo 1973. Edited by A. Hattori. Univ. Tokyo Press (Tokyo), 1975. MR 0372868 Zbl 0315.57003

Akio Hattori	Related
Robion C. Kirby	Related	Volume

M. G. Scharlemann and L. C. Siebenmann: “The Hauptvermutung for smooth singular homeomorphisms,” pp. 85–91 in Manifolds — Tokyo 1973 (Tokyo, 10–17 April 1973). Edited by A. Hattori. University of Tokyo Press, 1975. A part II (with somewhat different title) was published in Compositio Math. 29:3 (1974). MR 372871 Zbl 0315.57009 incollection

Laurent C. Siebenmann	Related
Akio Hattori	Related

M. Scharlemann: “Equivalence of 5-dimensional \( s \)-cobordisms,” Proc. Amer. Math. Soc. 53 : 2 (December 1975), pp. 508–510. MR 380838 Zbl 0285.57020 article

M. Scharlemann: “Constructing strange manifolds with the dodecahedral space,” Duke Math. J. 43 : 1 (March 1976), pp. 33–40. MR 402760 Zbl 0331.57007 article

Since its discovery by Poincaré at the beginning of the century, the dodecahedral manifold \( K \) has fueled several of the most fundamental theorems on manifolds. Originally \( K \) was presented as an example of a homology 3-sphere which is not a sphere, providing a counterexample to the “homology” Poincaré conjecture [Siefert and Threlfall 1934]. Milnor’s original counterexample \( \Sigma \) to the smooth Poincaré conjecture is closely connected to \( K \) in the theory of complex singularities [Milnor 1956; 1968]. Since \( \Sigma \) is a \( \mathit{PL} \) sphere, it provided the first example of a PL manifold with more than one smooth structure, thus distinguishing between the categories \( \mathit{DIFF} \) and \( \mathit{PL} \).

Most recently the manifold \( K \) has been used by Kirby and Siebenmann to distinguish between \( \mathit{PL} \) manifolds and merely topological manifolds. In particular, their fundamental example of a non-\( \mathit{PL} \) manifold is a 5-manifold \( M \) homotopy equivalent to \( X\times S^1 \), where \( X \) is a homology 4-manifold whose only non-Euclidean point has link \( K \) [Siebenmann 1970].

In this paper three manifolds are constructed which are closely related to the results of Kirby–Siebenmann. Their existence has been demonstrated elsewhere [Shaneson 1970], [Cappell and Shaneson 1971], [Hollingsworth and Morgan 1970]. Here the constructions flow from properties of \( K \). §1 presents two of the many descriptions of \( K \). In §2 fake homotopy structures are constructed for \[ S^3\times S^1 \mathbin{\#} S^2\times S^2 \quad\text{and}\quad S^3\times S^1\times S^1 .\] In §3 a nontriangulable 5-manifold homotopy equivalent to \( \mathit{CP}(2)\times S^1 \) is constructed, using deep results of Kirby–Siebenmann.

M. G. Scharlemann: “Transversality theories at dimension four,” Invent. Math. 33 : 1 (February 1976), pp. 1–14. MR 410756 Zbl 0318.57007 article

M. Scharlemann: “Simplicial triangulation of noncombinatorial manifolds of dimension less than 9,” Trans. Amer. Math. Soc. 219 (May 1976), pp. 269–287. MR 415629 Zbl 0333.57009 article

M. Scharlemann: “The fundamental group of fibered knot cobordisms,” Math. Ann. 225 : 3 (October 1977), pp. 243–251. MR 428338 Zbl 0325.55001 article

M. Scharlemann: “Thoroughly knotted homology spheres,” Houston J. Math. 3 : 2 (1977), pp. 271–283. MR 442947 Zbl 0355.57011 article

M. Scharlemann: “Isotopy and cobordism of homology spheres in spheres,” J. London Math. Soc. (2) 16 : 3 (December 1977), pp. 559–567. MR 464246 Zbl 0375.57003 article

Let \( H \) be a PL homology sphere of dimension \( m\geq 3 \). We intend to examine the degree to which the problem of classifying locally flat imbeddings \( H\to S^n \), \( n\geq 5 \), up to isotopy, is more complicated than that of classifying imbeddings \( S^m\to S^n \) (knot theory).

Here is an outline of the theory. There is a natural injection of knots into the set of imbeddings of \( H \), produced by adding knots to a canonical “primitive” imbedding of \( H \). Those imbeddings \( f:H\to S^n \) not in the image of this injection (called thorougly knotted imbeddings) are precisely those in which the inclusion induced map \[ \pi_1(H)\to \pi_1(S^n-f(H)) \] is non-trivial. In particular, this injection is always a bijection (i.e. there are no thoroughly knotted imbeddings of \( H \)) if \( n-m\geq 3 \). There are, however, homology spheres for which thoroughly knotted imbeddings exist if \( n-m\leq 2 \) [7]. In these codimensions, then, classifying imbeddings of homology spheres is more complex than classifying knots.

M. Scharlemann: “Non-PL imbeddings of 3-manifolds,” Amer. J. Math. 100 : 3 (June 1978), pp. 539–545. MR 515651 Zbl 0386.57009 article

All imbeddings will be locally flat and all isotopies will be ambient isotopies. Let \( M^m \), \( N^{m+2} \) be closed PL manifolds, \( m\geq 3 \). For any imbedding \[ f:M^m\to N^{m+2} \] there is an obstruction in \( H^3(M;\mathbb{Z}_2) \) to isotoping \( f \) to a PL imbedding [Rourke and Sanderson 1970]. However, it follows from the twisted product structure theorem of [Scharlemann and Siebenmann 1975] that for \( m\geq 5 \) there is a PL manifold \( M^{\prime} \) and a homeomorphism \( g:M^{\prime}\to M \) such that \[ f\circ g: M^{\prime}\to N \] is isotopic to a PL imbedding. Thus for \( m > 5 \) there is a natural way of replacing any imbedding of \( M \) in \( N \) with a PL imbedding of a homeomorphic manifold \( M^{\prime} \).

Here we study the analogous situation for \( m=3 \). Since PL structures on 3-manifolds are unique, the above theorem does not extend directly. However, we show there is a biunique correspondence between isotopy classes of non-PL imbeddings of \( M \) in \( N \) and isotopy classes of PL imbeddings (with an appropriate condition on fundamental group) of a manifold \( M^{\prime} \) homology equivalent to \( M \).

M. Scharlemann: “Smooth CE maps and smooth homeomorphisms,” pp. 234–240 in Algebraic and geometric topology: Proceedings of a symposium held at Santa Barbara in honor of Raymond L. Wilder (Santa Barbara, CA, 25–29 July 1977). Edited by K. C. Millett. Lecture Notes in Mathematics 664. Springer (Berlin), 1978. MR 518417 Zbl 0392.57002 incollection

Kenneth C. Millett	Related
Raymond Louis Wilder	Related

M. Scharlemann: “Transverse Whitehead triangulations,” Pac. J. Math. 80 : 1 (September 1979), pp. 245–251. MR 534713 Zbl 0419.57003 article

Suppose \( M \) and \( N \) are \( \mathit{PL} \) manifolds and \( f:M\to N \) is a proper \( \mathit{PL} \) map. Triangulate \( M \) and \( N \) so that \( f \) is simplical and let \( X \) be the dual complex in \( N \). Then for each open simplex \( \sigma \) in \( X \), \( f^{-1}(\sigma) \) is a \( \mathit{PL} \) submanifold of \( M \), so the stratification of \( N \) by the open simplices of \( X \) pulls back to a stratification of \( M \). In other words, any such \( \mathit{PL} \) map can be regarded as a map of combinatorially stratified sets in which each \( n \)-stratum of the range is a disjoint union of copies of \( R^n \). Here we prove the analogous theorem for a smooth map \( f:M\to N \) between smooth manifolds.

R. C. Kirby and M. G. Scharlemann: “Eight faces of the Poincaré homology 3-sphere,” pp. 113–146 in Geometric topology (Athens, GA, 1977). Edited by J. C. Cantrell. Academic Press (New York), 1979. MR 537730 Zbl 0469.57006

James Cecil Cantrell	Related
Robion C. Kirby	Related	Volume

M. Scharlemann: “Approximating \( \mathit{CAT} \) CE maps by \( \mathit{CAT} \) homeomorphisms,” pp. 475–501 in Geometric topology (Athens, GA, 1–12 August 1977). Edited by J. C. Cantrell. Academic Press (New York and London), 1979. MR 537746 Zbl 0474.57010 incollection

M. Scharlemann: “Subgroups of \( \mathrm{SL}(2,\mathbf{R}) \) freely generated by three parabolic elements,” Linear and Multilinear Algebra 7 : 3 (1979), pp. 177–191. MR 540952 Zbl 0412.20033 article

M. Scharlemann: “The subgroup of \( \Delta_{2} \) generated by automorphisms of tori,” Math. Ann. 251 : 3 (1980), pp. 263–268. MR 589255 article

M. Scharlemann: “The complex of curves on nonorientable surfaces,” J. London Math. Soc. (2) 25 : 1 (February 1982), pp. 171–184. MR 645874 Zbl 0479.57005 article

R. C. Kirby and M. G. Scharlemann: “Eight faces of the Poincaré homology 3-sphere,” Uspekhi Mat. Nauk 37 : 5(227) (1982), pp. 139–159. MR 676615 Zbl 0511.57008

M. Scharlemann: “Essential tori in 4-manifold boundaries,” Pac. J. Math. 105 : 2 (October 1983), pp. 439–447. MR 691614 Zbl 0516.57007 article

M. Scharlemann and C. Squier: “Automorphisms of the free group of rank two without finite orbits,” pp. 341–346 in Low-dimensional topology (San Francisco, CA, 7–11 January 1981). Edited by S. J. Lomonaco. Contemporary Mathematics 20. American Mathematical Society (Providence, RI), 1983. MR 718151 Zbl 0523.57002 incollection

Samuel James Lomonaco, Jr.	Related
Craig Cecil Squier	Related

M. Scharlemann: “The four-dimensional Schoenflies conjecture is true for genus two imbeddings,” Topology 23 : 2 (1984), pp. 211–217. MR 744851 Zbl 0543.57011 article

M. Scharlemann: “Tunnel number one knots satisfy the Poenaru conjecture,” Topology Appl. 18 : 2–3 (December 1984), pp. 235–258. MR 769294 Zbl 0592.57004 article

M. Scharlemann: “Smooth spheres in \( \mathbf{R}^4 \) with four critical points are standard,” Invent. Math. 79 : 1 (February 1985), pp. 125–141. MR 774532 Zbl 0559.57019 article

Let \( M \) be a smoothly imbedded 2-sphere in \( \mathbb{R}^4 \) on which some projection \( \mathbb{R}\to\mathbb{R} \) has four non-degenerate critical points. Here we show that \( M \) is isotopic to the standard 2-sphere in \( \mathbb{R}^4 \). This solves a question asked by Kuiper [1980]. The proof is based on a theorem first claimed by Hosokawa [1968], but with a major error (p. 253 1.3). This theorem also gives affirmative answers to questions 1.1 and 1.2A of [Kirby 1978], hence a negative answer to 1.3. The original solution to 1.1, proposed by Lickorish, uses deep results of Howie or Thurston and Gerstenhaber–Rothaus. In contrast the proof here is elementary in the sense that the argument is entirely self-contained and combinatorial. Indeed, the argument was available 50 years ago.

M. G. Scharlemann: “Unknotting number one knots are prime,” Invent. Math. 82 : 1 (February 1985), pp. 37–55. MR 808108 Zbl 0576.57004 article

M. Scharlemann: “Outermost forks and a theorem of Jaco,” pp. 189–193 in Combinatorial methods in topology and algebraic geometry (Rochester, NY, 29 June–2 July 1982). Edited by J. R. Harper and R. Mandelbaum. Contemporary Mathematics 44. American Mathematical Society (Providence, RI), 1985. Conference in honor of Arthur M. Stone. MR 813113 Zbl 0589.57011 incollection

Arthur Harold Stone	Related
John R. Harper	Related
Richard Mandelbaum	Related

M. Scharlemann: “3-manifolds with \( H_2(A,\partial A)=0 \) and a conjecture of Stallings,” pp. 138–145 in Knot theory and manifolds (Vancouver, BC, 2–4 June 1983). Edited by D. Rolfsen. Lecture Notes in Mathematics 1144. Springer (Berlin), 1985. MR 823287 Zbl 0585.57011 incollection

S. Bleiler and M. Scharlemann: “Tangles, property \( P \), and a problem of J. Martin,” Math. Ann. 273 : 2 (June 1986), pp. 215–225. MR 817877 Zbl 0563.57002 article

In [Scharlemann 1985a; 1985b] new combinatorial techniques are introduced to study the topology of planar surfaces in the complement of genus 2 handlebodies. Here we apply these techniques to two further problems. First, in an argument combining elements of [Scharlemann 1985a; 1985b], we solve problem 1.18B of [Kirby 1978]. The solution is obtained by precisely determining when a pair of unknots can arise during a single “move” in the Conway calculus. Secondly, we show that there is at most one non-trivial Dehn (i.e. rational) surgery on a strongly invertible knot that produces a simply-connected 3-manifold and that such a surgery must have coefficient \( \pm 1 \). This result has since been extended to all knots by Culler et al. [1987] using much less elementary methods. In a related, but combinatorially much more difficult paper [Bleiler and Scharlemann 1988] we solve problem 1.2B of [Kirby 1978] thereby eliminating even the coefficients \( \pm 1 \), and proving property \( P \) for strongly invertible knots.

M. Scharlemann: “A remark on companionship and property P,” Proc. Amer. Math. Soc. 98 : 1 (1986), pp. 169–170. MR 848897 Zbl 0602.57006 article

M. Scharlemann: “The Thurston norm and 2-handle addition,” Proc. Amer. Math. Soc. 100 : 2 (June 1987), pp. 362–366. MR 884480 Zbl 0627.57010 article

M. Scharlemann and A. Thompson: “Finding disjoint Seifert surfaces,” Bull. London Math. Soc. 20 : 1 (January 1988), pp. 61–64. MR 916076 Zbl 0654.57005 article

Given two Seifert surfaces \( S \) and \( T \) for a knot \( K \), there is a sequence of Seifert surfaces \( S = S_0 \), \( S_1,\dots \), \( S_n=T \) such that for each \( i \), \( 1\leq i\leq n \), \( S_i \) is disjoint from \( S_{i-1} \). The standard proof (see, for example [4]), which is useful in showing that any two Seifert matrices of \( K \) are \( S \)-equivalent, puts no limit on the genus of the intermediate Seifert surfaces \( S_1,\dots \), \( S_{n-1} \). Here we present a simple proof that

if \( S \) and \( T \) are of minimal genus, then we may take all \( S_i \) to be of minimal genus, and
for \( S \) an arbitrary Seifert surface, there is a sequence \( S = S_0 \), \( S_1,\dots \), \( S_n \) or Seifert surfaces such that \[ \operatorname{genus}(S_{i-1}) > \operatorname{genus}(S_i) ,\] \( S_n \) is of minimal genus, and for each \( i \), \( 1\leq i \leq n \), \[ S_i \cap S_{i-1} = \emptyset .\]

M. Scharlemann and A. Thompson: “Unknotting number, genus, and companion tori,” Math. Ann. 280 : 2 (March 1988), pp. 191–205. MR 929535 Zbl 0616.57003 article

In [Scharlemann 1985a] a complicated combinatorial argument showed that the band sum of knots is unknotted if and only if the band sum is a connected sum of unknots. This argument has since been dramatically simplified [Thompson 1987] and extended [Gabai 1987; S3, Sect. 8] using the newly developed machinery of Gabai. In [Scharlemann 1985b] a similar but more complicated combinatorial argument demonstrated that unknotting number one knots are prime. It seems natural to ask whether the Gabai machinery can simplify the proof of this old conjecture as well.

In fact the Gabai machine reveals a connection between the unknotting number of a knot, its genus, and the position of its companion tori. In Sect. 3 we show (roughly) that, if a single crossing change made to a knot \( K \) reduces its genus by more than one, then any companion torus to \( K \) can be made disjoint from the crossing. In particular, of \( K \) were composite and of unknotting number one, then the swallow-follow companion torus would remain as a companion to the unknot, which is impossible. Therefore no composite knot has unknotting number one.

This argument exploits the drop (by at least two) in the genus of a composite knot when a crossing change unknots it. It is natural to ask whether, in general, the genus of a knot drops (or at least does not rise) as it is unknotted by crossing changes. Knots exist for which a crossing change both lowers the unknotting number and raises the genus. A specific example (due to Chuck Livingston) is given in the appendix. Boileau and Murakami have shown us others. In Sect. 1 we give a general construction, again using the Gabai machine, which seems to produce myriads of examples.

Section 2 is a technical section which readies the Gabai machine (as presented in [Scharlemann 1989]) for use in Sect. 3. In Sect. 4 we view crossing changes as a special case of a more general operation, that of attaching an \( n \)-half-twisted band, and discuss how the main results of Sect. 3 generalize.

S. Bleiler and M. Scharlemann: “A projective plane in \( \mathbb{R}^4 \) with three critical points is standard. Strongly invertible knots have property \( P \),” Topology 27 : 4 (1988), pp. 519–540. MR 976593 Zbl 0678.57003 article

Let \( \mathbb{P} \) be the projective plane in \( \mathbb{R}^4 \) obtained by capping off the boundary of an unknotted Möbius band in \( \mathbb{R}^3\times\{0\} \) with an unknotted disk in \( \mathbb{R}^3\times [0,\infty) \). Here we show that any smoothly imbedded projective plane in \( \mathbb{R}^4 \) on which some projection \( \mathbb{R}^4\to\mathbb{R} \) has three nondegenerate critical points is isotopic to \( \mathbb{P} \). The proof is based on a combinatorial solution to Problem 1.2B of [Kirby 1978]. In particular, if a band is attached to an unknot so that the result is an unknot, then the band is isotopic to the trivial half-twisted band. One consequence is that strongly invertible knots have property \( P \) (see [Bleiler and Scharlemann 1986]). Together with [Culler et al. 1987], this further implies that pretzel knots (indeed all symmetric knots) have property \( P \).

M. Scharlemann and A. Thompson: “Link genus and the Conway moves,” Comment. Math. Helv. 64 : 4 (1989), pp. 527–535. MR 1022995 Zbl 0693.57004 article

M. Scharlemann: “Sutured manifolds and generalized Thurston norms,” J. Diff. Geom. 29 : 3 (1989), pp. 557–614. MR 992331 Zbl 0673.57015 article

M. Scharlemann: “Producing reducible 3-manifolds by surgery on a knot,” Topology 29 : 4 (1990), pp. 481–500. MR 1071370 Zbl 0727.57015 article

It has long been conjectured that surgery on a knot in \( S^3 \) yields a reducible 3-manifold if and only if the knot is cabled, with the cabling annulus part of the reducing sphere (cf. [Gordon 1983; Gordon and Litherland 1984; Gordon and Luecke 1987; Gonzalez-Acuña and Short 1986; Hatcher and Thurston 1985]). One may regard the Poenaru conjecture (solved in [Gabai 1987]) as a special case of the above. More generally, one can ask when surgery on a knot in an arbitary 3-manifold \( M \) produces a reducible 3-manifold \( M^{\prime} \). But this problem is too complex, since, dually, it asks which knots in which manifolds arise from surgery on reducible 3-manifolds. In this paper we are able to show, approximately, that if \( M \) itself either contains a summand not a rational homology sphere or is \( \partial \)-reducible, and \( M^{\prime} \) is reducible, then \( k \) must have been cabled and the surgery is via the slope of the cabling annulus. Thus the result stops short of proving the conjecture for \( M = S^3 \), but does suffice to prove the conjecture for satellite knots.

M. G. Scharlemann: “Lectures on the theory of sutured 3-manifolds,” pp. 25–45 in Algebra and topology 1990 (Taejon, S. Korea, 8–11 August 1990). Edited by S. H. Bae and G. T. Jin. Korea Advanced Institute of Science and Technology (Taejon), 1990. MR 1098719 Zbl 0756.57007 incollection

Sunghan Bae	Related
Gyo Taek Jin	Related

M. Scharlemann and A. Thompson: “Detecting unknotted graphs in 3-space,” J. Diff. Geom. 34 : 2 (1991), pp. 539–560. MR 1131443 Zbl 0751.05033 article

M. Scharlemann: “Handlebody complements in the 3-sphere: A remark on a theorem of Fox,” Proc. Amer. Math. Soc. 115 : 4 (August 1992), pp. 1115–1117. MR 1116272 Zbl 0759.57012 article

M. Scharlemann: “Some pictorial remarks on Suzuki’s Brunnian graph,” pp. 351–354 in Topology ’90 (Columbus, OH, February–June 1990). Edited by B. Apanasov, W. D. Neumann, A. W. Reid, and L. Siebenmann. Ohio State University Mathematics Research Institute Publications 1. de Gruyter (Berlin), 1992. MR 1184420 Zbl 0772.57005 incollection

Laurent C. Siebenmann	Related
Alan William Reid	Related
Walter D. Neumann	Related	Volume
Boris Nikolaevich Apanasov	Related

M. Scharlemann: “Topology of knots,” pp. 65–82 in Topological aspects of the dynamics of fluids and plasmas (Santa Barbara, CA, August–December 1991). Edited by H. K. Moffatt, G. M. Zaslavsky, P. Comte, and M. Tabor. NATO ASI Series. Series E. Applied Science 218. Kluwer Academic (Dordrecht), 1992. MR 1232225 Zbl 0799.57003 incollection

Pierre Comte	Related
Michael Tabor	Related
Henry Keith Moffatt	Related
George M. Zaslavsky	Related

M. Scharlemann: “Unlinking via simultaneous crossing changes,” Trans. Amer. Math. Soc. 336 : 2 (1993), pp. 855–868. MR 1200011 Zbl 0785.57003 article

M. Scharlemann and A. Thompson: “Heegaard splittings of \( (\textrm{surface})\times I \) are standard,” Math. Ann. 295 : 3 (1993), pp. 549–564. MR 1204837 Zbl 0814.57010 article

Frohman and Hass have shown [1989] that genus three Heegaard splittings of the 3-torus are standard. Boileau and Otal [1990] generalize this result to show that all Heegaard splittings of the 3-torus are standard. A crucial part of Boileau–Otal’s argument is to show that all Heegaard splittings of a torus crossed with an interval are standard. We generalize this part of their paper to prove that all Heegaard splittings of a closed orientable genus \( g \) surface crossed with an interval are standard. Many of our arguments are based on theirs; we differ substantially in Sect. 4, which allows us to obtain the more general result.

The paper is organized as follows: Section 1 begins with a discussion of compression bodies and their spines. In Sect. 2 we discuss Heegaard splittings and state the main theorem. The proof of the main theorem begins in Sect. 3, where we prove a lemma which splits the remainder of the proof into two cases. These cases are considered in Sect. 4 and Sect. 5. In Sect. 6 we exploit the main theorem to generalize a theorem of Frohman.

M. Scharlemann and Y. Q. Wu: “Hyperbolic manifolds and degenerating handle additions,” J. Austral. Math. Soc. Ser. A 55 : 1 (August 1993), pp. 72–89. MR 1231695 Zbl 0802.57005 article

A 2-handle addition on the boundary of a hyperbolic 3-manifold \( M \) is called degenerating if the resulting manifold is not hyperbolic. There are examples that some manifolds admit infinitely many degenerating handle additions. But most of them are not ‘basic’. (See Section 1 for definitions). Our first main theorem shows that there are only finitely many basic degenerating handle additions. We also study the case that one of the handle additions produces a reducible manifold, and another produces a \( \partial \)-reducible manifold, showing that in this case either the two attaching curves are disjoint, or they can be isotoped into a once-punctured torus. A byproduct is a combinatorial proof of a similar known result about degenerating hyperbolic structures by Dehn filling.

M. Scharlemann and A. Thompson: “Thin position and Heegaard splittings of the 3-sphere,” J. Diff. Geom. 39 : 2 (1994), pp. 343–357. MR 1267894 Zbl 0820.57005 article

We present here a simplified proof of the theorem, originally due to Waldhausen [1968], that a Heegaard splitting of \( S^3 \) is determined solely by its genus. The proof combines Gabai’s powerful idea of “thin position” [1987] with Johannson’s [1991] elementary proof of Haken’s theorem [1968] (Heegaard splittings of reducible 3-manifolds are reducible). In §3.1, 3.2 & 3.8 we borrow from Otal [1991] the idea of viewing the Heegaard splitting as a graph in 3-space in which we seek an unknotted cycle.

Along the way we show also that Heegaard splittings of boundary reducible 3-manifolds are boundary reducible [Casson and Gordon 1987, 1.2], obtain some (apparently new) characterizations of graphs in 3-space with boundary-reducible complement, and recapture a critical lemma of [Menasco and Thompson 1989].

M. Scharlemann and A. Thompson: “Thin position for 3-manifolds,” pp. 231–238 in Geometric topology (Haifa, Israel, 10–16 June 1992). Edited by C. Gordon, Y. Moriah, and B. Wajnryb. Contemporary Mathematics 164. American Mathematical Society (Providence, RI), 1994. MR 1282766 Zbl 0818.57013 incollection

Cameron McAllan Gordon	Related	Volume
Abigail A. Thompson	Related	Volume
Yoav Moriah	Related
Dov Bronislaw Wajnryb	Related

M. Scharlemann and A. Thompson: “Pushing arcs and graphs around in handlebodies,” pp. 163–171 in Low-dimensional topology. Edited by K. Johannson. Conference Proceedings and Lecture Notes in Geometry and Topology 3. International Press (Cambridge, MA), 1994. MR 1316180 Zbl 0868.57024 incollection

Abigail A. Thompson	Related	Volume
Klaus Johannson	Related

A. Bart and M. Scharlemann: “Least weight injective surfaces are fundamental,” Topology Appl. 69 : 3 (April 1996), pp. 251–264. MR 1382295 Zbl 0858.57016 article

H. Rubinstein and M. Scharlemann: “Comparing Heegaard splittings of non-Haken 3-manifolds,” Topology 35 : 4 (October 1996), pp. 1005–1026. MR 1404921 Zbl 0858.57020 article

Cerf theory can be used to compare two strongly irreducible Heegaard splittings of the same closed orientable 3-manifold. Any two splitting surfaces can be isotoped so that they intersect in a non-empty collection of curves, each of which is essential in both splitting surfaces. More generally, there are interesting isotopies of the splitting surfaces during which this intersection property is preserved. As sample applications we give new proofs of Waldhausen’s theorem that Heegaard splittings of \( \mathbb{S}^3 \) are standard, and of Bonahon and Otal’s theorem that Heegaard splittings of lens spaces are standard. We also present a solution to the stabilization problem for irreducible non-Haken 3-manifolds: If \( p \leq q \) are the genera of two splittings of such a manifold, then there is a common stabilization of genus \( 5p + 8q - 9 \).

H. Rubinstein and M. Scharlemann: “Transverse Heegaard splittings,” Mich. Math. J. 44 : 1 (1997), pp. 69–83. MR 1439669 Zbl 0907.57013 article

M. Scharlemann: “Planar graphs, family trees and braids,” pp. 29–47 in Progress in knot theory and related topics (Marseilles). Edited by M. Boileau, M. Domergue, Y. Mathieu, and K. Millett. Travaux en Cours 56. Hermann (Paris), 1997. MR 1603122 Zbl 0924.57002 incollection

Yves Mathieu	Related
Kenneth C. Millett	Related
Michel Charles Boileau	Related
Michel Domergue	Related

H. Rubinstein and M. Scharlemann: “Comparing Heegaard splittings: The bounded case,” Trans. Am. Math. Soc. 350 : 2 (1998), pp. 689–715. MR 1401528 Zbl 0892.57009 article

M. Scharlemann: “Crossing changes,” Chaos Solitons Fractals 9 : 4–5 (April–May 1998), pp. 693–704. Knot theory and its applications. MR 1628751 Zbl 0937.57004 article

M. Scharlemann: “Local detection of strongly irreducible Heegaard splittings,” Topology Appl. 90 : 1–3 (December 1998), pp. 135–147. MR 1648310 Zbl 0926.57018 article

M. Scharlemann and J. Schultens: “The tunnel number of the sum of \( n \) knots is at least \( n \),” Topology 38 : 2 (March 1999), pp. 265–270. MR 1660345 Zbl 0929.57003 article

D. Cooper and M. Scharlemann: “The structure of a solvmanifold’s Heegaard splittings,” pp. 1–18 in Proceedings of 6th Gökova geometry-topology conference (Gökova, Turkey, 25–29 May 1998), published as Turkish J. Math. 23 : 1. Issue edited by S. Akbulut, T. Önder, and R. J. Stern. Scientific and Technical Research Council of Turkey (Ankara), 1999. Dedicated to Rob Kirby on the occasion of his 60th birthday. MR 1701636 Zbl 0948.57015 incollection

Turgut Önder	Related
Robion C. Kirby	Related	Volume
Selman Akbulut	Related
Daryl Cooper	Related
Ronald John Stern	Related

Proceedings of the Kirbyfest (Berkeley, CA, June 22–26, 1998). Edited by J. Hass and M. Scharlemann. Geometry & Topology Monographs 2. Geometry & Topology Publications (Coventry), 1999. MR 1734398

H. Rubinstein and M. Scharlemann: “Genus two Heegaard splittings of orientable three-manifolds,” pp. 489–553 in Proceedings of the KirbyFest (Berkeley, CA, 22–26 June 1998). Edited by J. Hass and M. Scharlemann. Geometry & Topology Monographs 2. International Press (Cambridge, MA), 1999. Papers dedicated to Rob Kirby on the occasion of his 60th birthday. MR 1734422 Zbl 0962.57013 ArXiv 9712262 incollection

It was shown by Bonahon–Otal and Hodgson–Rubinstein that any two genus-one Heegaard splittings of the same 3-manifold (typically a lens space) are isotopic. On the other hand, it was shown by Boileau, Collins and Zieschang that certain Seifert manifolds have distinct genus-two Heegaard splittings. In an earlier paper, we presented a technique for comparing Heegaard splittings of the same manifold and, using this technique, derived the uniqueness theorem for lens space splittings as a simple corollary. Here we use a similar technique to examine, in general, ways in which two non-isotopic genus-two Heegard splittings of the same 3-manifold compare, with a particular focus on how the corresponding hyperelliptic involutions are related

J. Hyam Rubinstein	Related	Volume
Joel Hass	Related	Volume
Robion C. Kirby	Related	Volume

M. Scharlemann and J. Schultens: “Annuli in generalized Heegaard splittings and degeneration of tunnel number,” Math. Ann. 317 : 4 (2000), pp. 783–820. MR 1777119 Zbl 0953.57002 article

H. Goda, M. Scharlemann, and A. Thompson: “Levelling an unknotting tunnel,” Geom. Topol. 4 (2000), pp. 243–275. MR 1778174 Zbl 0958.57007 ArXiv math/9910099 article

Abigail A. Thompson	Related	Volume
Hiroshi Goda	Related

M. Scharlemann and J. Schultens: “Comparing Heegaard and JSJ structures of orientable 3-manifolds,” Trans. Amer. Math. Soc. 353 : 2 (2001), pp. 557–584. MR 1804508 Zbl 0959.57010 article

M. Jones and M. Scharlemann: “How a strongly irreducible Heegaard splitting intersects a handlebody,” Topology Appl. 110 : 3 (March 2001), pp. 289–301. MR 1807469 Zbl 0974.57011 article

M. Scharlemann: “Heegaard reducing spheres for the 3-sphere,” Rend. Istit. Mat. Univ. Trieste 32 : supplement 1 (2001), pp. 397–410. Dedicated to the memory of Marco Reni. MR 1893407 Zbl 1030.57031 article

M. Scharlemann: “The Goda–Teragaito conjecture: An overview,” pp. 87–102 in On Heegaard splittings and Dehn surgeries of 3-manifolds, and topics related to them (Kyoto, 11–15 June 2001), published as RIMS Kōkyūroku 1229. Issue edited by T. Kobayashi. 2001. MR 1905564 incollection

M. Scharlemann: “Heegaard splittings of compact 3-manifolds,” pp. 921–953 in Handbook of geometric topology. Edited by R. J. Daverman and R. B. Sher. North-Holland (Amsterdam), 2002. MR 1886684 Zbl 0985.57005 ArXiv math/0007144 incollection

Richard Benjamin Sher	Related
Robert J. Daverman	Related

M. Scharlemann and A. Thompson: “Unknotting tunnels and Seifert surfaces,” Proc. London Math. Soc. (3) 87 : 2 (2003), pp. 523–544. MR 1990938 Zbl 1047.57008 ArXiv math/0010212 article

Let \( K \) be a knot with an unknotting tunnel \( \gamma \) and suppose that \( K \) is not a 2-bridge knot. There is an invariant \[ \rho = p/q\in\mathbb{Q}/2\mathbb{Z} ,\] with \( p \) odd, defined for the pair \( (K,\gamma) \).

The invariant \( \rho \) has interesting geometric properties. It is often straightforward to calculate; for example, for \( K \) a torus knot and \( \gamma \) an annulus-spanning arc, \[ \rho(K,\gamma) = 1 .\] Although \( \rho \) is defined abstractly, it is naturally revealed when \( K\cup\gamma \) is put in thin position. If \( \rho\neq 1 \) then there is a minimal-genus Seifert surface \( F \) for \( K \) such that the tunnel \( \gamma \) can be slid and isotoped to lie on \( F \). One consequence is that if \[ \rho(K,\gamma)\neq 1 \] then \( K > 1 \). This confirms a conjecture of Goda and Teragaito for pairs \( (K,\gamma) \) with \[ \rho(K,\gamma)\neq 1 .\]

M. Scharlemann and A. Thompson: “Thinning genus two Heegaard spines in \( S^3 \),” J. Knot Theor. Ramif. 12 : 5 (2003), pp. 683–708. MR 1999638 Zbl 1048.57002 article

M. Scharlemann: “Heegaard splittings of 3-manifolds,” pp. 25–39 in Low dimensional topology: Lectures at the Morningside Center of Mathematics (Beijing, 1998–1999). Edited by B. Li, S. Wang, and X. Zhao. New Studies in Advanced Mathematics 3. International Press (Somerville, MA), 2003. MR 2052244 Zbl 1044.57006 incollection

Shicheng Wang	Related
Benghe Li	Related
Xuezhi Zhao	Related

M. Scharlemann: “There are no unexpected tunnel number one knots of genus one,” Trans. Amer. Math. Soc. 356 : 4 (2004), pp. 1385–1442. MR 2034312 Zbl 1042.57003 article

M. Scharlemann and A. Thompson: “On the additivity of knot width,” pp. 135–144 in Proceedings of the Casson Fest (Fayetteville, AR, 10–12 April 2003 and Austin, TX, 19–21 May 2003). Edited by C. Gordon and Y. Rieck. Geometry & Topology Monographs 7. Geometry & Topology Publications (Coventry, UK), 2004. Based on the 28th University of Arkansas spring lecture series in the mathematical sciences and a conference on the topology of manifolds of dimensions 3 and 4. This paper was “Dedicated to Andrew Casson, a mathematician’s mathematician.”. MR 2172481 Zbl 1207.57016 ArXiv math/0403326 incollection

Andrew J. Casson	Related
Abigail A. Thompson	Related	Volume
Yo'av Rieck	Related
Cameron McAllan Gordon	Related	Volume

@incollection {key2172481m,
	AUTHOR = {Scharlemann, Martin and Thompson, Abigail},
	TITLE = {On the additivity of knot width},
	BOOKTITLE = {Proceedings of the {C}asson {F}est},
	EDITOR = {Gordon, Cameron and Rieck, Yoav},
	SERIES = {Geometry \& Topology Monographs},
	NUMBER = {7},
	PUBLISHER = {Geometry \& Topology Publications},
	ADDRESS = {Coventry, UK},
	YEAR = {2004},
	PAGES = {135--144},
	DOI = {10.2140/gtm.2004.7.135},
	NOTE = {(Fayetteville, AR, 10--12 April 2003
		 and Austin, TX, 19--21 May 2003). Based
		 on the 28th University of Arkansas spring
		 lecture series in the mathematical sciences
		 and a conference on the topology of
		 manifolds of dimensions 3 and 4. This
		 paper was ``Dedicated to Andrew Casson,
		 a mathematician's mathematician''. ArXiv:math/0403326.
		 MR:2172481. Zbl:1207.57016.},
	ISSN = {1464-8989},
}

M. Scharlemann: “Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting,” Bol. Soc. Mat. Mexicana (3) 10 : special issue (2004), pp. 503–514. MR 2199366 Zbl 1095.57017 ArXiv math/0307231 article

M. Scharlemann and A. Thompson: “Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken 3-manifolds,” Proc. Am. Math. Soc. 133 : 6 (2005), pp. 1573–1580. MR 2120271 Zbl 1071.57015 ArXiv math/0308011 article

M. Scharlemann: “Thin position in the theory of classical knots,” Chapter 9, pp. 429–459 in Handbook of knot theory. Edited by W. Menasco and M. Thistlethwaite. Elsevier (Amsterdam), 2005. MR 2179267 Zbl 1097.57013 incollection

Morwen Bernard Thistlethwaite	Related
William Wyatt Menasco	Related

M. Scharlemann and J. Schultens: “3-manifolds with planar presentations and the width of satellite knots,” Trans. Amer. Math. Soc. 358 : 9 (2006), pp. 3781–3805. MR 2218999 Zbl 1102.57004 article

M. Scharlemann and M. Tomova: “Alternate Heegaard genus bounds distance,” Geom. Topol. 10 (2006), pp. 593–617. MR 2224466 Zbl 1128.57022 article

M. Scharlemann: “Proximity in the curve complex: Boundary reduction and bicompressible surfaces,” Pac. J. Math. 228 : 2 (December 2006), pp. 325–348. MR 2274524 Zbl 1127.57010 article

Suppose \( N \) is a compressible boundary component of a compact irreducible orientable 3-manifold \( M \), and \[ (Q,\partial Q)\subset (M,\partial M) \] is an orientable properly embedded essential surface in \( M \), some essential component of which is incident to \( N \) and no component is a disk. Let \( \mathscr{V} \) and \( \mathscr{Q} \) denote respectively the sets of vertices in the curve complex for \( N \) represented by boundaries of compressing disks and by boundary components of \( Q \). We prove that, if \( Q \) is essential in \( M \), then \[ d(\mathscr{V},\mathscr{Q})\leq 1 -\chi(Q) .\]

Hartshorn showed that an incompressible surface in a closed 3-manifold puts a limit on the distance of any Heegaard splitting. An augmented version of our result leads to a version of Hartshorn’s theorem for merely compact 3-manifolds.

Our main result is: If a properly embedded connected surface \( Q \) is incident to \( N \), and \( Q \) is separating and compresses on both its sides, but not by way of disjoint disks, then either \[ d(\mathscr{V},\mathscr{Q})\leq 1 -\chi(Q) ,\] or \( Q \) is obtained from two nested connected incompressible boundary-parallel surfaces by a vertical tubing.

Forthcoming work with M. Tomova will show how an augmented version of this theorem leads to the same conclusion as Hartshorn’s theorem, not from an essential surface, but from an alternate Heegaard surface. That is, if \( Q \) is a Heegaard splitting of a compact \( M \) then no other Heegaard splitting has distance greater than twice the genus of \( Q \).

M. Scharlemann: “Generalized property \( R \) and the Schoenflies conjecture,” Comment. Math. Helv. 83 : 2 (2008), pp. 421–449. MR 2390052 Zbl 1148.57032 article

M. Scharlemann and M. Tomova: “Uniqueness of bridge surfaces for 2-bridge knots,” Math. Proc. Cambridge Philos. Soc. 144 : 3 (May 2008), pp. 639–650. MR 2418708 Zbl 1152.57006 article

M. Scharlemann and M. Tomova: “Conway products and links with multiple bridge surfaces,” Mich. Math. J. 56 : 1 (2008), pp. 113–144. MR 2433660 Zbl 1158.57011 article

A link \( K \) in a 3-manifold \( M \) is said to be in bridge position with respect to a Heegaard surface \( P \) for \( M \) if each arc of \( K-P \) is parallel to \( P \), in which case \( P \) is called a bridge surface for \( K \) in \( M \). Given a link in bridge position with respect to \( P \), it is easy to construct more complex bridge surfaces for \( K \) from \( P \) — for example, by stabilizing the Heegaard surface \( P \) or by perturbing \( K \) to introduce a minimum and an adjacent maximum. As with Heegaard splitting surfaces for a manifold, it is likely that most links have multiple bridge surfaces even apart from these simple operations. In an effort to understand how two bridge surfaces for the same link might compare, it seems reasonable to follow the program used in [Rubinstein and Scharlemann 1997] to compare distinct Heegaard splittings of the same non-Haken 3-manifold. The restriction to non-Haken manifolds ensured that the relevant Heegaard splittings were strongly irreducible. In our context the analogous condition is that the bridge surfaces are c-weakly incompressible (definition to follow). The natural analogy to the first step in [Rubinstein and Scharlemann 1997] would be to demonstrate that any two distinct c-weakly incompressible bridge surfaces for a link \( K \) in a closed orientable 3-manifold \( M \) can be isotoped so that their intersection consists of a nonempty collection of curves, each of which is essential (including nonmeridional) on both surfaces. In some sense the similar result in [Rubinstein and Scharlemann 1997] could then be thought of as the special case in which \( K=\emptyset \).

Here we demonstrate that this is true when there are no incompressible Conway spheres for the knot \( K \) in \( M \) (cf. Section 4 and [Gordon and Luecke 2006]). In the presence of Conway spheres a slightly different outcome cannot be ruled out: the bridge surfaces each intersect a collar of a Conway sphere in a precise way; outside the collar the bridge surfaces intersect only in curves that are essential on both surfaces; and inside the collar there is inevitably a single circle intersection that is essential in one surface and meridional — and hence inessential — in the other.

The Zieschang Gedenkschrift: A memorial volume for Heiner Zieschang (1936–2004). Edited by M. Boileau, M. Scharlemann, and R. Weidmann. Geometry and Topology Monographs 14. Geometry & Topology Publications (Coventry, UK), 2008. MR 2484694 Zbl 1135.00012 book

Richard Weidmann	Related
Heiner Zieschang	Related
Michel Charles Boileau	Related

M. Scharlemann: “Refilling meridians in a genus 2 handlebody complement,” pp. 451–475 in The Zieschang Gedenkschrift: A memorial volume for Heiner Zieschang (1936–2004). Edited by M. Boileau, M. Scharlemann, and R. Weidmann. Geometry and Topology Monographs 14. Geometry & Topology Publications (Coventry, UK), 2008. Dedicated to the memory of Heiner Zieschang, first to notice that genus two handlebodies could be interesting. MR 2484713 Zbl 1177.57019 incollection

Richard Weidmann	Related
Heiner Zieschang	Related
Michel Charles Boileau	Related

M. Scharlemann and A. A. Thompson: “Surgery on a knot in (surface \( \times I \)),” Algebr. Geom. Topol. 9 : 3 (2009), pp. 1825–1835. MR 2550096 Zbl 1197.57011 ArXiv 0807.0405 article

Suppose \( F \) is a compact orientable surface, \( K \) is a knot in \( F\times I \), and \( (F\times I)_{\textrm{surg}} \) is the 3-manifold obtained by some nontrivial surgery on \( K \). If \( F\times\{0\} \) compresses in \( (F\times I)_{\textrm{surg}} \), then there is an annulus in \( F\times I \) with one end \( K \) and the other end an essential simple closed curve in \( F\times\{0\} \). Moreover, the end of the annulus at \( K \) determines the surgery slope.

An application: Suppose \( M \) is a compact orientable 3-manifold that fibers over the circle. If surgery on \( K\subset M \) yields a reducible manifold, then either

the projection \( K\subset M\to S^1 \) has nontrivial winding number,
\( K \) lies in a ball,
\( K \) lies in a fiber, or
\( K \) is cabled.

R. Qiu and M. Scharlemann: “A proof of the Gordon conjecture,” Adv. Math. 222 : 6 (December 2009), pp. 2085–2106. MR 2562775 Zbl 1180.57025 article

R. E. Gompf, M. Scharlemann, and A. Thompson: “Fibered knots and potential counterexamples to the property \( 2{R} \) and slice-ribbon conjectures,” Geom. Topol. 14 : 4 (2010), pp. 2305–2347. MR 2740649 Zbl 1214.57008 ArXiv 1103.1601 article

If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions.

The simplest plausible counterexample to the Generalized Property R Conjecture could be a 2-component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to \( \#_2(S^1\times S^2) \). We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that are not known to be ribbon.

Robert E. Gompf	Related
Abigail A. Thompson	Related	Volume

M. Scharlemann: “An overview of Property 2R,” pp. 317–325 in The mathematics of knots. Edited by M. Banagl and D. Vogel. Contributions in Mathematical and Computational Sciences 1. Springer (Berlin), 2011. MR 2777854 Zbl 1221.57012 incollection

The celebrated Property R Conjecture, affirmed by David Gabai (J. Differ. Geom. 26:461, 1987), can be viewed as the first stage of a sequence of conjectures culminating in what has been called the Generalized Property R Conjecture. This conjecture is relevant to the study of outstanding problems in both 3-manifolds (specifically, links in \( S^3 \)) and 4-manifolds (specifically, the Schoenflies Conjecture and the smooth Poincaré Conjecture). Here we give an overview of part of forthcoming work of R. Gompf, A. Thompson and the author which considers the next stage in such a progression, called the Property 2R Conjecture.

It is shown that the lowest genus counterexample (if any exists) cannot be fibered. Exploiting Andrews–Curtis type considerations on presentations of the trivial group, it is argued that one of the simplest possible candidates for a counterexample, the square knot, probably is one. This suggests there is a genus one counterexample, though we have so far been unable to identify it. Finally, we note that the counterexample need not be an obstacle to the sort of 4-manifold consequences towards which the Generalized Property R Conjecture is aimed.

Denis Vogel	Related
Markus Banagl	Related

J. Berge and M. Scharlemann: “Multiple genus 2 Heegaard splittings: A missed case,” Algebr. Geom. Topol. 11 : 3 (2011), pp. 1781–1792. MR 2821441 Zbl 1232.57020 article

M. Scharlemann: “Berge’s distance 3 pairs of genus 2 Heegaard splittings,” Math. Proc. Cambridge Philos. Soc. 151 : 2 (September 2011), pp. 293–306. MR 2823137 Zbl 1226.57033 article

M. Scharlemann: “Generating the genus \( g+1 \) Goeritz group of a genus \( g \) handlebody,” pp. 347–369 in Geometry and topology down under (Melbourne, 11–22 July 2011). Edited by C. D. Hodgson, W. H. Jaco, M. G. Scharlemann, and S. Tillmann. Contemporary Mathematics 597. American Mathematical Society (Providence, RI), 2013. MR 3186683 Zbl 1288.57014 incollection

William H. Jaco	Related
Craig David Hodgson	Related
Stephan Tillmann	Related

Geometry and topology down under: A conference in honour of Hyam Rubinstein (Melbourne, Australia, 11–22 July 2011). Edited by C. D. Hodgson, W. H. Jaco, M. G. Scharlemann, and S. Tillmann. Contemporary Mathematics 597. American Mathematical Society (Providence, RI), 2013. MR 3202515 Zbl 1272.57002 book

J. Hyam Rubinstein	Related	Volume
William H. Jaco	Related
Craig David Hodgson	Related
Stephan Tillmann	Related

M. Scharlemann, J. Schultens, and T. Saito: Lecture notes on generalized Heegaard splittings (Kyoto, 11–15 June 2001). World Scientific (Hackensack, NJ), 2016. Three lectures on low-dimensional topology in Kyoto. MR 3585907 Zbl 1356.57004 book

Jennifer Carol Schultens	Related
Toshio Saito	Related

M. Scharlemann: “Proposed Property 2R counterexamples examined,” Ill. J. Math. 60 : 1 (2016), pp. 207–250. To Wolfgang Haken, who, forty years ago, found that Four Colors Suffice. MR 3665179 Zbl 1376.57012 article

In 1985, Akbulut and Kirby analyzed a homotopy 4-sphere \( \Sigma \) that was first discovered by Cappell and Shaneson, depicting it as a potential counterexample to three important conjectures, all of which remain unresolved. In 1991, Gompf’s further analysis showed that \( \Sigma \) was one of an infinite collection of examples, all of which were (sadly) the standard \( S^4 \), but with an unusual handle structure.

Recent work with Gompf and Thompson, showed that the construction gives rise to a family \( L_n \) of 2-component links, each of which remains a potential counterexample to the generalized Property R Conjecture. In each \( L_n \), one component is the simple square knot \( Q \), and it was argued that the other component, after handle-slides, could in theory be placed very symmetrically. How to accomplish this was unknown, and that question is resolved here, in part by finding a symmetric construction of the \( L_n \). In view of the continuing interest and potential importance of the Cappell–Shaneson–Akbulut–Kirby–Gompf examples (e.g., the original \( \Sigma \) is known to embed very efficiently in \( S^4 \) and so provides unique insight into proposed approaches to the Schoenflies Conjecture) digressions into various aspects of this view are also included.

M. Freedman and M. Scharlemann: Powell moves and the Goeritz group. Preprint, 2018. ArXiv 1804.05909 techreport

M. Freedman and M. Scharlemann: “Dehn’s lemma for immersed loops,” Math. Res. Lett. 25 : 6 (2018), pp. 1827–1836. MR 3934846 article

M. Scharlemann: A strong Haken’s Theorem. Preprint, 2020. ArXiv 2003.08523 techreport

M. Freedman and M. Scharlemann: Uniqueness in Haken’s Theorem. Preprint, 2020. ArXiv 2004.07385 techreport

Martin Scharlemann

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