Celebratio Mathematica

[1] J. W. Morgan: Stable tangential homotopy equivalances. Ph.D. thesis, Rice University (Ann Arbor, MI), 1969. MR 2618330 Zbl 0176.22006 phdthesis

[2] N. Levitt and J. W. Morgan: “Transversality structures and p.l. structures on spherical fibrations,” Bull. Amer. Math. Soc. 78 (1972), pp. 1064–1068. MR 314050 Zbl 0267.55021 article

[3] G. W. Brumfiel and J. W. Morgan: “Quadratic functions, the index modulo 8, and a \( \mathbf{ Z}/4 \)-Hirzebruch formula,” Topology 12 (1973), pp. 105–122. MR 324709 Zbl 0254.57007 article

[4] J. W. Morgan and D. P. Sullivan: “The transversality characteristic class and linking cycles in surgery theory,” Ann. of Math. (2) 99 (1974), pp. 463–544. MR 350748 Zbl 0295.57008 article

[5] J. Morgan and R. Scott: “A nodal basis for \( C^{1} \) piecewise polynomials of degree \( n\geq 5 \),” Math. Comput. 29 (1975), pp. 736–740. MR 375740 Zbl 0307.65074 article

[6] P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan: “Real homotopy theory of Kähler manifolds,” Invent. Math. 29 : 3 (1975), pp. 245–274. MR 382702 Zbl 0312.55011 article

[7] G. W. Brumfiel and J. W. Morgan: “Homotopy theoretic consequences of N. Levitt’s obstruction theory to transversality for spherical fibrations,” Pacific J. Math. 67 : 1 (1976), pp. 1–100. MR 431185 Zbl 0343.55019 article

The main goal of this paper is a detailed analysis of the problem of imposing a topological bundle structure on a spherical fibre space over a simply connected base. The method involves a careful study of the notion of fibre homotopy transversality due to N. Levitt. The point is, a topological disc bundle satisfies strong transversality properties for maps from manifolds to the associated Thorn space. These properties can be formulated at least for spherical fibre spaces. Thus, obstructions to transversality can be interpreted as obstructions to imposing a topological bundle structure on a spherical fibre space. It turns out that over a simply connected base the obstructions to transversality coincide exactly with the obstructions to a topological structure.

The obstructions to transversality for a spherical fibre space \( \xi \) can be interpreted as obstructions to a deformation of the identity map on the Thom space \( T\xi \) to a certain subcomplexes \( W\xi \). The fibre of the map \( W\xi \to T\xi \) is a space with a suitable iterated loop space homotopy equivalent to \( G/TOP \). The total obstruction to transversality becomes the obstruction to a \( KO \otimes \mathbb{Z}[1/2] \) orientation of the Thom space \( T\xi \), mixed with certain cohomology classes of \( T\xi \), \( \tilde{\mathcal{L}} \in H^{4^{\ast} +1}(T\xi, \mathbb{Z}_{(2)}) \) and \( \tilde{\mathcal{K}}\in H^{4^{\ast} -1} (T\xi , \mathbb{Z}/2) \). These obstructions are then also interpretable as the obstructions to lifting in the fibration sequence \( G/TOP \to BSTOP \to BSG \).

[8] F. Griffits, P. Delin’, D. Morgan, and D. Sjullivan: “Real homotopy theory of Kähler manifolds,” Uspehi Mat. Nauk 32 : 3(195) (1977), pp. 119–152, 247. In Russian. Translated from the English by Ju. I. Manin. Translation of English original in Invent. Math. 29: 3 (1975), 245–274. MR 460700 Zbl 0355.55016 article

[9] J. W. Morgan: “The rational homotopy theory of smooth, complex projective varieties (following P. Deligne, P. Griffiths, J. Morgan, and D. Sullivan) (Invent. Math. 29 (1975), no. 3, 245–274),” pp. [exposé] 475, pp. 69–80 in Séminaire Bourbaki, 1975–76. Lecture Notes in Math. 567. Springer (Berlin), 1977. MR 454967 Zbl 0361.32009 incollection

The results on the rational homotopy types of smooth, projective varieties (or more generally Kähler manifolds) were motivated by Sullivan’s theory relating the differential forms on a manifold to its homotopy type. Kähler manifolds present themselves as an example where interesting and highly nontrivial properties hold for the differential forms. Here we examine the homotopy theoretic consequences of these properties. We begin by recounting Sullivan’s theory — first the general theory of homotopy types of differential algebras, then a little rational homotopy theory for spaces, and finally the connection between the two. After this we develop some of the properties of forms on a Kähler manifold and deduce the results.

[10] J. W. Morgan: A product formula for surgery obstructions, vol. 14. Mem. Amer. Math. Soc. 201. American Mathematical Society (Providence, RI), 1978. MR 501030 Zbl 0388.57017 book

Given a degree one normal map and a closed manifold one can form their product. The result is another degree one normal map. In this paper we calculate the surgery obstruction of the product in terms of the obstruction of the original normal map and invariants of the closed manifold provided that the latter is simply connected. The result depends on the congruence class modulo 4 of the dimension of the closed manifold. Let us call that dimension \( d \). If \( d \) is congruent to 2 or 3 modulo 4 then the resulting product has zero surgery obstruction. If \( d \) is congruent to 0 modulo 4 then the obstruction of the product is the original obstruction multiplied by the index of the closed manifold. If \( d \) is congruent to 1 modulo 4 then the obstruction of the product depends only on the original obstruction and on the de Rham invariant of the closed manifold. We give an example of the last type where the resulting product obstruction is non zero.

[11] J. W. Morgan: “The algebraic topology of smooth algebraic varieties,” Inst. Hautes Études Sci. Publ. Math. 48 (1978), pp. 137–204. MR 516917 Zbl 0401.14003 article

[12] J. W. Morgan: “Hodge theory for the algebraic topology of smooth algebraic varieties,” pp. 119–127 in Algebraic and geometric topology (Proc. Sympos. Pure Math.) (Stanford Univ., Stanford, CA, 1976). Edited by R. J. Milgram. Proc. Sympos. Pure Math. 32. American Mathematical Society (Providence, RI), 1978. In 2 volumes. MR 520528 Zbl 0417.14005 incollection

The Hodge theory of harmonic forms, when applied to smooth, complex projective varieties, gives deep and important results about the cohomology of such manifolds. In this article I wish to show that the same Hodge theory applies equally well to give nontrivial statements about algebraic topological invariants beyond cohomology. The results will be valid for any smooth, complex, algebraic variety (not just projective ones). That part of the algebraic topology which is amenable to study via Hodge theory is the rational algebraic topology. Since Hodge theory is based on differential forms, all subtle torsion and divisibility questions are ignored. One might be tempted to say that it is possible to study only real (i.e., over \( R \)) algebraic topology via forms, but since all real algebraic topology of spaces carries an inherent rational structure, the results that we obtain descend automatically from \( R \) to \( Q \).

[13] J. W. Morgan: “Nonsingular Morse–Smale flows on 3-dimensional manifolds,” Topology 18 : 1 (1979), pp. 41–53. MR 528235 Zbl 0406.58020 article

In this paper we study which compact, orientable 3-manifolds \( W \) have non-singular Morse–Smale flows which are transverse to \( \partial W \) and pointing inward exactly on \( \partial_{W} \) (\( \partial_{W} \) is a union of components of \( \partial W \)). There is an obvious necessary condition — namely that \( \chi ( W, \partial_W) = 0 \). If \( W \) has dimension different from 3, then Asimov [1, 2] showed that \( ( W, \partial_W) \) does indeed have a non-singular Morse–Smale flow provided that the Euler characteristic condition is satisfied. His method was to compare Morse–Smale flows with what he called round handle decompositions, and then to prove that manifolds satisfying the Euler characteristic condition admit round handle decompositions. Unfortunately, his argument could not be adapted to work in dimension 3. Here, we shall show that the result which Asimov obtained in all other dimensions is not true in dimension 3.

[14] M. Davis, W. C. Hsiang, and J. W. Morgan: “Concordance classes of regular \( \mathrm{ O}(n) \)-actions on homotopy spheres,” Acta Math. 144 : 3–4 (1980), pp. 153–221. MR 573451 Zbl 0453.57026 article

[15] P. A. Griffiths and J. W. Morgan: Rational homotopy theory and differential forms. Progress in Mathematics 16. Birkhäuser (Boston), 1981. MR 641551 Zbl 0474.55001 book

[16] J. Carlson, H. Clemens, and J. Morgan: “On the mixed Hodge structure associated to \( \pi_{3} \) of a simply connected complex projective manifold,” Ann. Sci. École Norm. Sup. (4) 14 : 3 (1981), pp. 323–338. MR 644521 Zbl 0511.14005 article

[17] J. W. Morgan: “Actions de groupes finis sur \( S^3 \): La conjecture de P. A. Smith (d’après Thurston et Meeks–Yau),” pp. [exposé] 578, 277 in Bourbaki Seminar, 1980–81, vol. 1980/81, Exposés 561–578. Lecture Notes in Math. 901. Springer (Berlin), 1981. MR 647502 incollection

[18] E. Cavazzuti and J. Morgan: “Problèmes d’optimisation uniformément bien posés et méthodes de pénalisation,” Boll. Un. Mat. Ital. B (6) 1 : 2 (1982), pp. 423–450. MR 666579 Zbl 0499.49018 article

[19] J. W. Morgan: “Topological triviality of various analytic families,” Duke Math. J. 50 : 1 (1983), pp. 215–225. MR 700138 Zbl 0543.14010 article

[20] J. W. Morgan: “The Smith conjecture,” pp. 3–6 in The Smith conjecture (New York, 1979). Edited by J. W. Morgan and H. Bass. Pure Appl. Math. 112. Academic Press (Orlando, FL), 1984. MR 758460 Zbl 0599.57001 incollection

This chapter discusses the Smith conjecture. The proof of the Smith conjecture represents a culmination of the efforts of many mathematicians. The work of Smith on cyclic group actions was seminal. The chapter also describes the formulations and generalizations of the Smith conjecture. The arguments proving the Smith conjecture can easily be adapted for the piecewise linear (PL) case or for the topological case, provided that in the topological case it is assumed that the fixed point set is locally flat. The techniques used to establish the Smith conjecture can be used to prove various generalizations. The theorem on the solution of the Smith conjecture affirms several special cases of the Poincaré conjecture. The applications of these ideas and methods to three-dimensional topology are due to Thurston, Meeks, and Yau, with help from Bass, Shalen, Gordon, and Litherland.

[21] J. W. Morgan: “An outline of the proof,” pp. 11–16 in The Smith conjecture (New York, 1979). Edited by J. W. Morgan and H. Bass. Pure Appl. Math. 112. Academic Press (Orlando, FL), 1984. MR 758462 Zbl 0990.49029 incollection

The chapter presents an outline of the proof of the Smith conjecture with illustrations of theorems. Making some reductions, the chapter proceeds to sketch an argument to complete the proof with the help of a few case studies. The techniques are those of classical piecewise-linear (PL) topology, plus an essential new fact — namely, the equivariant version of Dehn’s lemma and the loop theorem. The hypotheses of one of these cases are exactly those of Thurston’s uniformization theorem, which argues that \( \Sigma - K \) admits a hyperbolic structure. This representation is automatically irreducible. At this point, the argument becomes purely algebraic. A flow chart shown in the chapter gives a pictorial representation of the logical structure of the argument in support of the proof.

[22] J. W. Morgan: “History of the Smith conjecture and early progress,” pp. 7–9 in The Smith conjecture (New York, 1979). Edited by J. W. Morgan and H. Bass. Pure Appl. Math. 112. Academic Press (Orlando, FL), 1984. MR 758461 incollection

This chapter discusses the history of the Smith conjecture and early progress. The study of periodic diffeomorphisms of the disk and the sphere began with the work of Brouwer and Kerékjártó in the 1920s. They proved that an orientation-preserving periodic diffeomorphism of the 2-disk or the 2-sphere was conjugate by a diffeomorphism to a rotation. These original proofs were incomplete and the gap was filled later by Eilenberg. Smith studied homeomorphisms of the \( n \)-disk and the \( n \)-sphere periodic of prime power period. He proved that the \( Z/p \)-homology of the fixed point set is the same as that of a smaller dimensional disk or sphere. When the homeomorphism is orientation-preserving, then the codimension of the fixed point set is even. The analogue of the Smith conjecture in dimension \( m > 3 \) was investigated by Giffen, who showed that for each \( n > 1 \), \( m > 4 \) there exists a smooth \( m \)-sphere pair that is inequivalent to the standard pair.

[23] J. W. Morgan: “On Thurston’s uniformization theorem for three-dimensional manifolds,” pp. 37–125 in The Smith conjecture (New York, 1979). Edited by J. W. Morgan and H. Bass. Pure Appl. Math. 112. Academic Press (Orlando, FL), 1984. MR 758464 Zbl 0599.57002 incollection

This chapter presents an outline of Thurston’s existence theorem for hyperbolic structures on three-dimensional manifolds — the three-dimensional uniformization theorem. It is based on lectures given by Thurston at the symposium on the Smith conjecture. The chapter provides a complete proof of the uniformization theorem. The criterion for deciding to treat the various pieces was simple: the parts of the argument that are more formal rely on the Bers–Ahlfors–Teichmüller theory or rely on three-dimensional topology. The statements of the main theorem concerning the existence of hyperbolic structures in the case when the manifolds have finite volume are considered. The chapter describes all the topological results necessary to carry out the inductive process of cutting apart the three-manifolds. The chapter focuses on Maskit’s combination theorems that deal with gluing Kleinian groups together along common quasi-Fuchsian subgroups.

[24] J. W. Morgan and P. B. Shalen: “Valuations, trees, and degenerations of hyperbolic structures, I,” Ann. of Math. (2) 120 : 3 (1984), pp. 401–476. MR 769158 Zbl 0583.57005 article

The theory developed in this paper and its sequel [MS1] was motivated by the feeling that what was done in [CS] for curves in the algebraic set \( X \) of \( SL_2(\mathbb{C}) \)-characters of a finitely generated group should have a natural generalization to arbitrary algebraic subvarieties of \( X \), and that such a generalization would be an important tool in the study of surfaces and 3-manifolds. The algebraic aspect of the theory is developed in this paper and the geometric-topological aspect in [MS1]. As applications we give proofs, from an entirely new point of view, of two fundamental results of Thurston’s: In Chapter III of this paper there is a description of a natural boundary for Teichmüller space, and in [MS1] there is a characterization of those compact 3-manifolds whose space of homotopy-hyperbolic structures is compact. Because our methods are drawn from the mathematical mainstream, and therefore help to explain Thurston’s results by putting them in a wider framework, there is hope that they will be more generally applicable.

[25] M. W. Davis and J. W. Morgan: “Finite group actions on homotopy 3-spheres,” pp. 181–225 in The Smith conjecture (New York, 1979). Edited by J. W. Morgan and H. Bass. Pure Appl. Math. 112. Academic Press (Orlando, FL), 1984. MR 758469 Zbl 0599.57008 incollection

The chapter discusses the finite group actions on homotopy 3-spheres. The Smith conjecture states that certain types of smooth actions of a cyclic group on a homotopy 3-sphere are essentially linear. This hypothesis is proved in a substantial number of cases based on theorems presented in the chapter. It is an immediate consequence of the Schönflies theorem and the definition of essentially linear that any essentially linear action on \( S^3 \) is equivariantly diffeomorphic to a linear action. The chapter introduces the notion of an orbifold coined by Thurston to describe spaces that are locally the quotients of finite group actions. The chapter discusses two- and three-dimensional examples of these spaces. The language of orbifolds is a convenient framework for dealing with the quotient spaces of properly discontinuous group actions. The connection between finite group actions and linear actions through the intermediary of Seifert-fibered orbifolds is established.

[26] J. W. Morgan and P. B. Shalen: “An introduction to compactifying spaces of hyperbolic structures by actions on trees,” pp. 228–240 in Geometry and topology (College Park, MD, 1983/84). Edited by J. Alexander and J. Harer. Lecture Notes in Math. 1167. Springer, 1985. MR 827272 Zbl 0592.57007 incollection

[27] J. W. Morgan: “Group actions on trees and the compactification of the space of classes of \( {\mathrm SO}(n,1) \)-representations,” Topology 25 : 1 (1986), pp. 1–33. MR 836721 Zbl 0595.57030 article

The purpose of this paper is to adapt the theory of [6] to study deformations of hyperbolic \( n \)-manifolds. In [6] a theory was developed to study deformations of hyperbolic surfaces and 3-manifolds. This theory was based on the fact that the groups of automorphisms of the hyperbolic plane and 3-space are respectively \( PSL_2 (\mathbb{R}) \) and \( PSL_2 (\mathbb{C}) \). Using a variant of the Serre-theory of the tree associated to \( SL_2 \), we were able to compactify the space of \( SL_2 \)-characters of a given group \( \Gamma \) and identify the points at infinity with certain actions of \( \Gamma \) on trees. Here, we work with the linear algebraic group \( SO (n, 1) \). The connection with hyperbolic geometry is that the component of the identity \( SO^+_{\mathbb{R}} (n, 1) \) of the real points of \( SO (n, 1) \) is identified with the group of orientation-preserving isometries of hyperbolic \( n \)-space.

[28] J. W. Morgan and I. Morrison: “A van Kampen theorem for weak joins,” Proc. London Math. Soc. (3) 53 : 3 (1986), pp. 562–576. MR 868459 Zbl 0609.57002 article

[29] J. W. Morgan: “Correction to: ‘The algebraic topology of smooth algebraic varieties’,” Inst. Hautes Études Sci. Publ. Math. 64 (1986), pp. 185. Corrections to the article published in {Inst. {H}autes Études {S}ci. {P}ubl. {M}ath.}, 48 (1978), 137–204. MR 876163 Zbl 0617.14013 article

[30] R. Friedman and J. W. Morgan: “On the diffeomorphism types of certain elliptic surfaces,” pp. 115–127 in Geometry and topology (Athens, GA, 1985). Edited by C. McCrory and T. Shifrin. Lecture Notes in Pure and Appl. Math. 105. Dekker (New York), 1987. MR 873289 Zbl 0611.57018 incollection

The purpose of this note is to describe some recent results concerning certain closed, smooth 4-manifolds. These 4-manifolds have the structure of algebraic surfaces (more precisely, they are simply connected elliptic surfaces). Our results depend on a new invariant introduced by Donaldson in [3]. He used this invariant to prove that two well-known elliptic surfaces, which are homotopy equivalent (even homeomorphic) and hence h-cobordant, are not diffeomorphic. By using the formal properties he established for this invariant, we are able to extend his results to an entire class of elliptic surfaces. We show, in fact, that there are infinitely many simply connected 4-manifolds, all homotopy equivalent to one another (and hence homeomorphic), no two of which are diffeomorphic. The same techniques may be applied to study the group of self-diffeomorphisms of these elliptic surfaces. We show that, in the group of integral automorphisms of the cohomology ring, the subgroup of elements realized by self-diffeomorphisms may be characterized up to finite index and is of infinite index.

[31] G. Baumslag, J. W. Morgan, and P. B. Shalen: “Generalized triangle groups,” Math. Proc. Cambridge Philos. Soc. 102 : 1 (1987), pp. 25–31. MR 886432 Zbl 0626.20023 article

[32] R. Friedman, B. Moishezon, and J. W. Morgan: “On the \( C^\infty \) invariance of the canonical classes of certain algebraic surfaces,” Bull. Amer. Math. Soc. (N.S.) 17 : 2 (1987), pp. 283–286. MR 903733 Zbl 0627.57014 article

The results announced in this article concern certain aspects of the diffeomorphism classification of algebraic surfaces, and in particular, the role of the canonical class. We establish our results by developing a general criterion under which the possibilities for Donaldson’s polynomial invariants for smooth 4-manifolds [2] are severely limited. We then use these limitations to conclude that in many cases the canonical class of an algebraic surface is a diffeomorphism invariant up to a multiple. Two classes of surfaces satisfying our general criterion are complete intersections and simply connected elliptic surfaces with \( pg \equiv 0\, (\mathrm{mod}\ 2) \) (see Corollary 8). A third class of such surfaces are certain abelian branched coverings of \( CP^1 \times CP^1 \) which are surfaces of general type (see §4). These latter surfaces provide infinitely many examples of pairs of homeomorphic, nondiffeomorphic, simply connected surfaces of general type.

§2 gives a brief review of the part of Donaldson’s theory needed for what we do here. The material described in §3 represents the work of the first and last authors and is some evidence for a general conjecture described in [5]. The material described in §4 represents the work of the second author and will be explained in detail in [6].

[33] M. Culler and J. W. Morgan: “Group actions on \( \mathbb{R} \)-trees,” Proc. London Math. Soc. (3) 55 : 3 (1987), pp. 571–604. MR 907233 Zbl 0658.20021 article

An \( \mathbb{R} \)-tree is a non-empty metric space in which any two points are joined by a unique arc, and in which every arc is isometric to a closed interval in the real line. Group actions on \( \mathbb{R} \)-trees arise naturally from groups of isometries of hyperbolic space, and have had significant application in the study of hyperbolic manifolds. In [6], [11] and [13] it is shown that the space of conjugacy classes of representations of a finitely generated group \( G \) into \( \mathrm{SO}(n, 1) \) has a natural compactification whose ideal points are isomorphism classes of actions of \( G \) on \( \mathbb{R} \)-trees. From a different point of view, both Gromov and Thurston [17] have constructed \( \mathbb{R} \)-trees as limits of sequences of hyperbolic spaces, scaled so that the curvature goes to \( -\infty \). These theorems suggest that the study of group actions on \( \mathbb{R} \)-trees can be viewed as a natural extension of representation theory; we take this point of view here.

There are two features of group actions on \( \mathbb{R} \)-trees which are addressed in this paper. The first is that isometries of \( \mathbb{R} \)-trees behave in many ways like isometries of hyperbolic space, and that groups of isometries of \( \mathbb{R} \)-trees resemble subgroups of \( \mathrm{SO}(n, 1) \). The second is that, for a fixed finitely generated group \( G \), the space of all actions of \( G \) on \( \mathbb{R} \)-trees has strong compactness properties.

[34] J. W. Morgan: “Trees and hyperbolic geometry,” pp. 590–597 in Proceedings of the International Congress of Mathematicians (Berkeley, CA, 1986). Edited by A. M. Gleason. American Mathematical Society (Providence, RI), 1987. In two volumes. MR 934260 Zbl 0681.57025 inproceedings

[35] R. Friedman and J. W. Morgan: “Algebraic surfaces and 4-manifolds: some conjectures and speculations,” Bull. Amer. Math. Soc. (N.S.) 18 : 1 (1988), pp. 1–19. MR 919651 Zbl 0662.57016 article

[36] R. Friedman and J. W. Morgan: “On the diffeomorphism types of certain algebraic surfaces, I,” J. Differential Geom. 27 : 2 (1988), pp. 297–369. MR 925124 Zbl 0669.57016 article

[37] J. W. Morgan and P. B. Shalen: “Degenerations of hyperbolic structures, II: Measured laminations in 3-manifolds,” Ann. of Math. (2) 127 : 2 (1988), pp. 403–456. MR 932305 Zbl 0656.57003 article

This is the second paper in a series of three. The first in the series was [7]. That paper concerned the general theory of groups acting on \( \mathbb{R} \)-trees and the relationship of these actions to representations into \( \mathrm{SL}_2(\mathrm{C}) \). The purpose of the present paper is to develop the foundations of a general theory of measured laminations in 3-manifolds. In the third paper [8] the results of this paper will be applied to study actions of 3-manifold groups on trees. (The starting point of [8] is the idea that “dual” to an action of \( \pi_1(M) \) on an \( \mathbb{R} \)-tree is a codimension-1 measured lamination in \( M \).) Combining the latter results with [7] gives information about how hyperbolic structures on 3-manifolds can degenerate.

[38] R. Friedman and J. W. Morgan: “On the diffeomorphism types of certain algebraic surfaces, II,” J. Differential Geom. 27 : 3 (1988), pp. 371–398. MR 940111 Zbl 0669.57017 article

[39] J. W. Morgan and P. B. Shalen: “Degenerations of hyperbolic structures, III: Actions of 3-manifold groups on trees and Thurston’s compactness theorem,” Ann. of Math. (2) 127 : 3 (1988), pp. 457–519. MR 942518 Zbl 0661.57004 article

[40] J. W. Morgan: “Ergodic theory and free actions of groups on \( \mathbf{ R} \)-trees,” Invent. Math. 94 : 3 (1988), pp. 605–622. MR 969245 Zbl 0676.57001 article

[41] R. Friedman and J. W. Morgan: “Complex versus differentiable classification of algebraic surfaces,” pp. 135–139 in Proceedings of the 1987 Georgia Topology Conference (Athens, GA, 1987), published as Topology Appl. 32 : 2. Issue edited by N. Habegger and C. McCrory. 1989. MR 1007985 Zbl 0694.14013 inproceedings

[42] F. A. Griffiths and J. W. Morgan: Ratsional’naya teoriya gomotopiĭ i differentsial’nye formy. Nauka (Moscow), 1990. Translated from the English and with an appendix by A. V. Pazhitnov. Russian translation of 1981 original. MR 1108177 book

[43]J. W. Morgan, T. Mrowka, and D. Ruberman: Self-intersection numbers of embedded 2-spheres in algebraic surfaces, 1991. Unpublished manuscript. misc

[44] M. Fernández, A. Gray, and J. W. Morgan: “Compact symplectic manifolds with free circle actions, and Massey products,” Michigan Math. J. 38 : 2 (1991), pp. 271–283. MR 1098863 Zbl 0726.53028 article

[45] J. W. Morgan and P. B. Shalen: “Free actions of surface groups on \( {\mathbb{R}} \)-trees,” Topology 30 : 2 (1991), pp. 143–154. MR 1098910 Zbl 0726.57001 article

In [9] Lyndon introduced a class of real-valued functions on groups, now called Lyndon length functions, and showed that a group is free if and only if it admits an integer-valued Lyndon length function. He raised the question of which groups admit \( \mathbb{R} \)-valued Lyndon length functions. Using the construction of Chiswell [3] this question can be re-interpreted as asking which groups act freely (by isometries) on \( \mathbb{R} \)-trees. (For the definition of an \( \mathbb{R} \)-tree see [12].)

The examples of such groups pointed out by Lyndon are arbitrary free products of subgroups of \( \mathbb{R} \). The first examples not of this type were given by Alperin Moss in [Z]. Their examples arc not finitely generated. In this paper we give the first finitely generated examples which are not free products of free abelian groups.

[46] J. W. Morgan: \( \Lambda \)-trees and their applications, 1991. Video recording of a lecture presented in Columbus, Ohio, August 1990, 1 videocassette (NTSC; 1/2 inch; VHS) (60 min.); sd., col. MR 1153665 Zbl 0925.20038 misc

[47] J. W. Morgan and R. K. Skora: “Groups acting freely on \( \mathbf{ R} \)-trees,” Ergodic Theory Dynam. Systems 11 : 4 (1991), pp. 737–756. MR 1145619 Zbl 0766.57021 article

In this paper we study the question of which groups act freely on \( \mathbb{R} \)-trees. The paper has two parts. The first part concerns groups which contain a non-cyclic, abelian subgroup. The following is the main result in this case.

Let the finitely presented group \( G \) act freely on an \( \mathbb{R} \)-tree. If \( A \) is a non-cyclic, abelian subgroup of \( G \), then \( A \) is contained in an abelian subgroup \( A^{\prime} \) which is a free factor of \( G \).

The second part of the paper concerns groups which split as an \( HNN \)-extension along an infinite cyclic group. Here is one formulation of our main result in that case.

Let the finitely presented group \( G \) act freely on an \( R \)-tree. If \( G \) has an \( HNN \)-decomposition \[ G= H\ast{\langle s\rangle}, \] where \( \langle s\rangle \) is infinite cyclic, then there is a subgroup \( H^{\prime} \subset H \) such that either

\( G=H^{\prime}\ast \mathbb{Z} \); or
\( G=H^{\prime} \ast \pi_1 S\ast \mathbb{Z}\ast\dots\ast \mathbb{Z} \), where \( S \) is a closed surface of non-positive Euler characteristic.

A slightly different, more precise result is also given.

[48] J. W. Morgan: “\( \Lambda \)-trees and their applications,” Bull. Amer. Math. Soc. (N.S.) 26 : 1 (1992), pp. 87–112. MR 1100579 Zbl 0767.05054 article

To most mathematicians and computer scientists the word “tree” conjures up, in addition to the usual image, the image of a connected graph with no circuits. We shall deal with various aspects and generalizations of these mathematical trees. (As Peter Shalen has pointed out, there will be leaves and foliations in this discussion, but they do not belong to the trees!) In the last few years various types of trees have been the subject of much investigation. But this activity has not been exposed much to the wider mathematical community. To me the subject is very appealing for it mixes very naïve geometric considerations with the very sophisticated geometric and algebraic structures. In fact, part of the drama of the subject is guessing what type of techniques will be appropriate for a given investigation: Will it be direct and simple notions related to schematic drawings of trees or will it be notions from the deepest parts of algebraic group theory, ergodic theory, or commutative algebra which must be brought to bear? Part of the beauty of the subject is that the naive tree considerations have an impact on these more sophisticated topics. In addition, trees form a bridge between these disparate subjects.

[49] J. W. Morgan and T. S. Mrowka: “A note on Donaldson’s polynomial invariants,” Internat. Math. Res. Notices 10 (1992), pp. 223–230. MR 1191573 Zbl 0787.57011 article

The purpose of this note is to exhibit two uses of the idea of stabilizing Donaldson polynomial invariants via the connected sum with \( \overline{CP}^2 \). The theme that we develop is that this operation allows one to avoid the anomalous behavior associated with flat connections. In some cases this simplifies arguments, and in other cases it actually permits one to extend definitions and computations which are not possible (at least directly) without the stabilization. Our first application concerns generalizing the usual \( SU(2) \)-Donaldson polynomial invariants. Consider the Donaldson polynomial invariants of \( M \# \overline{CP}^2 \) associated to \( U(2) \)-bundles with \( c_1 \) equal to the exceptional class \( [\overline{CP}^1] \). We show that these invariants are defined for any closed oriented smooth 4-manifold with \( b^+_2(M) > 1 \). These invariants are polynomials in the exceptional class with coefficients which are themselves polynomial invariants of \( M \). We show that, in the case when \( M \) is simply connected and the charge \( p_1 \) is sufficiently negative, the coefficient of the first power of the exceptional class agrees, up to a factor of \( (- 1)/2 \), with the usual \( SU(2) \)-Donaldson polynomial invariant of \( M \). In this way we are led (after inverting 2) to a generalization of the Donaldson polynomial invariants to invariants for any closed, oriented 4-manifold with \( b > 1 \). (See Corollary 2.2 and Proposition 3.2.) Our second application is to give a simpler proof of Donaldson’s connected-sum theorem. (See Theorem 3.1 and Corollary 3.6.) Our proof uses dimension counts exactly as in Donaldson’s original argument [3] for the case when the limit connections on the sides are not flat, but, by the use of \( U(2) \)-connections with nontrivial \( c_1 \), avoids the more subtle case when the connection on one of the sides is flat.

[50] J. W. Morgan and J.-P. Otal: “Relative growth rates of closed geodesics on a surface under varying hyperbolic structures,” Comment. Math. Helv. 68 : 2 (1993), pp. 171–208. MR 1214228 Zbl 0795.57009 article

[51] J. W. Morgan and T. S. Mrowka: “On the diffeomorphism classification of regular elliptic surfaces,” Internat. Math. Res. Notices 6 (1993), pp. 183–184. MR 1224116 Zbl 0807.57015 article

[52] J. W. Morgan: “Comparison of the Donaldson polynomial invariants with their algebro-geometric analogues,” Topology 32 : 3 (1993), pp. 449–488. MR 1231956 Zbl 0801.57014 article

[53] J. W. Morgan and K. G. O’Grady: Differential topology of complex surfaces: Elliptic surfaces with \( p_g=1 \): smooth classification. Lecture Notes in Mathematics 1545. Springer, 1993. With the collaboration of Millie Niss. MR 1312610 Zbl 0789.14037 book

This book is about the smooth classification of a certain class of algebraic surfaces, namely regular elliptic surfaces of geometric genus one, i.e., elliptic surfaces with \( b_1 = 0 \) and \( b_2^+ = 3 \). The authors give a complete classification of these surfaces up to diffeomorphism. They achieve this result by partially computing one of Donaldson’s polynomial invariants. The computation is carried out using techniques from algebraic geometry. In these computations both the basic facts about the Donaldson invariants and the relationship of the moduli space of ASD connections with the moduli space of stable bundles are assumed known. Some familiarity with the basic facts of the theory of moduli of sheaves and bundles on a surface is also assumed. This work gives a good and fairly comprehensive indication of how the methods of algebraic geometry can be used to compute Donaldson invariants.

[54] J. W. Morgan, T. Mrowka, and D. Ruberman: The \( L^2 \)-moduli space and a vanishing theorem for Donaldson polynomial invariants. Monographs in Geometry and Topology 2. International Press (Somerville, MA), 1994. MR 1287851 Zbl 0830.58005 book

[55] D. Kotschick and J. W. Morgan: “\( \mathrm{ SO}(3) \)-invariants for 4-manifolds with \( b^+_2=1 \), II,” J. Differential Geom. 39 : 2 (1994), pp. 433–456. MR 1267898 Zbl 0828.57013 article

In this paper we extend the definition of Donaldson polynomial invariants to cover the case of manifolds with \( b_1 = 0 \) and \( b^+_2 = 1 \). We shall consider \( SO(3) \)-bundles with \( w_2 \) which lifts to an integral class. This paper generalizes [2] where the case of \( SU(2) \)-bundles with \( c_2 = 1 \) was treated. It should be viewed as the continuation of [5], where \( SO(3) \)-bundles with arbitrary \( p_1 \) were considered. It extends [5] in two ways. First of all, it completes the proof of the fact that the values of these invariants depend only on the chamber containing the self-dual harmonic 2-form for the metric used to define the anti-self-dual (ASD) equation. Secondly, it establishes more of the general properties of the differences of the values of the invariants as the self-dual 2-form crosses a wall. It follows from the properties that we establish here that, as conjectured in [5], the value of an invariant on every chamber is determined by its value on any one chamber; and in particular, the invariant is defined for all chambers regardless of whether they contain forms which are self-dual harmonic for some metric.

[56] R. Friedman and J. W. Morgan: Smooth four-manifolds and complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 27. Springer, 1994. MR 1288304 Zbl 0817.14017 book

The heart of this work is a continuation of the study, via algebro-geometric techniques, of the Donaldson polynomial invariants of algebraic surfaces, and in particular, elliptic surfaces. The main result, which is proved by combining the classical algebro-geometric classification with these computations, is that there is only a finite discrepancy between the differentiable classification of complex algebraic surfaces and the complex analytic classification of algebraic surfaces up to deformation of their complex structure. (Two surfaces are of the same deformation type if there is a connected analytic family of surfaces that contains the two given surfaces. Roughly speaking this means that the two surfaces are diffeomorphic and that, using an appropriate diffeomorphism to identify them, we can find a continuous path of complex structures joining the two given ones.) More precisely, we shall prove the following two theorems:

Let \( M^4 \) be a smooth, oriented 4-manifold. Then the number of deformation equivalence classes of algebraic surfaces which are diffeomorphic to M is finite.

Theorem. Let \( S_1 \) and \( S_2 \) be two complex surfaces. If \( S_1 \) is diffeomorphic to \( S_2 \) , then either \( S_1 \) and \( S_2 \) have the same Kodaira dimension or one of the surfaces \( S_1 \) , \( S_2 \) is a rational surface and the other is a surface of general type.

[57] D. Kotschick, J. W. Morgan, and C. H. Taubes: “Four-manifolds without symplectic structures but with nontrivial Seiberg–Witten invariants,” Math. Res. Lett. 2 : 2 (1995), pp. 119–124. MR 1324695 Zbl 0853.57020 article

[58] J. W. Morgan and T. S. Mrowka: “The smooth classification of elliptic surfaces,” pp. 246–292 in Geometry, topology, & physics for Raoul Bott. Edited by S.-T. Yau. Conf. Proc. Lecture Notes Geom. Topology. International Press (Somerville, MA), 1995. MR 1358620 Zbl 0874.57020 incollection

[59] J. W. Morgan: The Seiberg–Witten equations and applications to the topology of smooth four-manifolds. Mathematical Notes 44. Princeton University Press (Princeton, NJ), 1996. MR 1367507 Zbl 0846.57001 book

[60] J. W. Morgan, Z. Szabó, and C. H. Taubes: “A product formula for the Seiberg–Witten invariants and the generalized Thom conjecture,” J. Differential Geom. 44 : 4 (1996), pp. 706–788. MR 1438191 Zbl 0974.53063 article

The Thom Conjecture asserts that any compact, embedded surface in \( CP^2 \) of degree \( d > 0 \) must have genus at least as large as the smooth algebraic curve of the same degree, namely \( (d - l)(d - 2)/2 \). More generally, one can ask whether in any algebraic surface a smooth algebraic curve is of minimal genus in its homology class. There was one significant result in this direction. Using \( SU(2) \)-Donaldson invariants, Kronheimer showed in [3] that this result is true for curves of positive self-intersection in a large class of simply connected surfaces with \( b^{+}_2 > 1 \). Unfortunately, for technical reasons, this argument does not extend to cover the case of \( CP^2 \). It is the purpose of this paper to prove the general result that a smooth holomorphic curve of non-negative self-intersection in a compact Kähler manifold is genus minimizing.

[61] R. Friedman, J. Morgan, and E. Witten: “Vector bundles and \( \mathrm{ F} \) theory,” Comm. Math. Phys. 187 : 3 (1997), pp. 679–743. MR 1468319 article

[62] R. Friedman and J. W. Morgan: “Algebraic surfaces and Seiberg–Witten invariants,” J. Algebraic Geom. 6 : 3 (1997), pp. 445–479. MR 1487223 article

[63] J. W. Morgan and Z. Szabó: “Homotopy \( K3 \) surfaces and mod 2 Seiberg–Witten invariants,” Math. Res. Lett. 4 : 1 (1997), pp. 17–21. MR 1432806 article

[64] J. W. Morgan, T. S. Mrowka, and Z. Szabó: “Product formulas along \( T^3 \) for Seiberg–Witten invariants,” Math. Res. Lett. 4 : 6 (1997), pp. 915–929. MR 1492130 article

[65] J. W. Morgan: “An introduction to gauge theory,” pp. 51–143 in Gauge theory and the topology of four-manifolds (Park City, UT, 1994). Edited by R. Friedman and J. W. Morgan. IAS/Park City Math. Ser. 4. American Mathematical Society (Providence, RI), 1998. MR 1612968 Zbl 0911.57024 incollection

Gauge theory is the study of principal bundles, connections on them, and the curvatures of these connections. Much of it is an analytic study of certain especially important types of connections — called (Anti)-Self-Dual connections. These objects were introduced by physicists and have proved to be a central ingredient in modern particle physics, being the mechanism that allows for unifying various of the four fundamental forces. The subject thus has many sides — the physical, differential geometric, analytic, and the topological being the most important. We are interested in the application of these techniques to the classification of four-dimensional smooth manifolds. Before turning our attention to these applications, however, we need to understand the other sides of the subject. In fact, most of this course will consist of a basic introduction to the analytic, topological, and differential geometric aspects of gauge theory. Only toward the end will we turn to its applications. Even then we will only be able to scratch the surface — a much more detailed study than we have time for is necessary before one can attempt more serious applications.

[66] R. Friedman, J. W. Morgan, and E. Witten: “Principal \( G \)-bundles over elliptic curves,” Math. Res. Lett. 5 : 1–2 (1998), pp. 97–118. MR 1618343 Zbl 0937.14019 article

Let \( E \) be an elliptic curve with origin \( p_0 \), and let \( G \) be a complex simple algebraic group. For simplicity, we shall only consider the case where \( G \) is simply connected, although all of the methods discussed below can be extended to the case of a general group \( G \). The goal of this note is to announce some results concerning the moduli of principal holomorphic \( G \)-bundles over \( E \). Detailed proofs, as well as a more thorough discussion of the case where \( E \) is allowed to be singular or to vary in families and of the connections with del Pezzo surfaces, elliptic \( K3 \) surfaces, and Calabi-Yau manifolds which are elliptic or \( K3 \) fibrations, will appear elsewhere.

[67] J. W. Morgan and Z. Szabó: “On \( h \)-cobordisms and Seiberg–Witten invariants,” pp. 117–124 in Topics in symplectic 4-manifolds (Irvine, CA, 1996). Edited by R. J. Stern. First Int. Press Lect. Ser. 1. International Press (Somerville, MA), 1998. MR 1635699 Zbl 0929.57021 incollection

[68] J. W. Morgan and Z. Szabó: “Embedded tori in four-manifolds,” Topology 38 : 3 (1999), pp. 479–496. MR 1670388 Zbl 0926.57023 article

In this paper we study how generalized logarithmic transformations and generalized fiber sums change the Donaldson invariant of smooth closed four manifolds. The two basic operations are defined as follows.

Let \( X \) be a smooth closed oriented four manifold, and \( L\hookrightarrow X \) be a smoothly embedded 2-torus with self-intersection 0. Let \( nd(L) \) denote a tubular neighborhood of \( L \). Then for each orientation reversing diffeomorphism \[ \phi : \partial(X\backslash nd (L)) \rightarrow \partial (D^2\times T^2) \] the resulting closed manifold \[ X(\phi) = (X\backslash nd (L))\cup_{\phi} (D^2\times T^2) \] is the generalized logarithmic transformation of \( X \) corresponding to \( \phi \).

Similarly, let \( Z_1 \) and \( Z_2 \) be smooth closed oriented four manifolds with smoothly embedded two-tori \( L_i\hookrightarrow Z_i \) with self-intersection 0, where \( i=1, 2 \). Then for each orientation reversing diffeomorphism \[ \phi:\partial (Z_1\backslash nd (L_1)) \rightarrow \partial (Z_2\backslash nd (L_2)) \] we define the generalized fiber sum \[ Z(\phi) = (Z_1 \backslash nd (L_1)) \cup_{\phi} (Z_2 \backslash nd (L_2)) .\]

Particular cases of these constructions give the classical log transform and fiber connected sum for elliptic surfaces.

[69] R. Friedman, J. W. Morgan, and E. Witten: “Vector bundles over elliptic fibrations,” J. Algebraic Geom. 8 : 2 (1999), pp. 279–401. MR 1675162 Zbl 0937.14004 article

[70] J. W. Morgan and Z. Szabó: “Complexity of 4-dimensional \( h \)-cobordisms,” Invent. Math. 136 : 2 (1999), pp. 273–286. MR 1688374 Zbl 0929.57022 article

[71] R. Friedman and J. W. Morgan: “Obstruction bundles, semiregularity, and Seiberg–Witten invariants,” Comm. Anal. Geom. 7 : 3 (1999), pp. 451–495. MR 1698386 Zbl 0946.14034 article

[72] P. Deligne and J. W. Morgan: “Notes on supersymmetry (following Joseph Bernstein),” pp. 41–97 in Quantum fields and strings: A course for mathematicians (Princeton, NJ, 1996/1997). Edited by P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison, and E. Witten. American Mathematical Society (Providence, RI), 1999. In 2 volumes. MR 1701597 Zbl 1170.58302 incollection

[73] J. W. Morgan: “Smooth invariants of 4-manifolds,” pp. 95–189 in Low dimensional topology. Edited by J. Böröczky, Károly, W. Neumann, and A. Stipsicz. Bolyai Soc. Math. Stud. 8. János Bolyai Math. Soc. (Budapest), 1999. Proceedings of five lecture series held during the Summer School on Low Dimensional Topology, 2–14 August 14, 1998 in Budapest, Hungary, and at the EMS Summer Schools No. 1, Algebraic Geometry, in Eger, Hungary in 1996. With discussion sessions by András I. Stipsicz. MR 1747269 Zbl 0946.57022 incollection

[74] J. W. Morgan: “Holomorphic bundles over elliptic manifolds,” pp. 135–203 in School on Algebraic Geometry (Trieste, Italy, 15 July–13 August 1999). Edited by L. Göttsche. ICTP Lect. Notes 1. Abdus Salam Int. Cent. Theoret. Phys. (Trieste), 2000. MR 1795863 Zbl 0995.32014 incollection

In this lecture we shall examine holomorphic bundles over compact elliptically fibered manifolds. We shall examine constructions of such bundles as well as (duality) relations between such bundles and other geometric objects, namely \( K \)3-surfaces and del Pezzo surfaces.

We shall be dealing throughout with holomorphic principal bundles with structure group \( G_{\mathrm{C}} \) where \( G \) is a compact, simple (usually simply connected) Lie group and \( G_{\mathrm{C}} \) is the associated complex simple algebraic group. Of course, in the special case \( G = SU(n) \) and hence \( G_{\mathrm{C}} = SL_n(\mathrm{C}) \), we are considering holomorphic vector bundles with trivial determinant. In the other cases of classical groups, \( G =SO(n) \) or \( G = \operatorname{Sympl}(2n) \) we are considering holomorphic vector bundles with trivial determinant equipped with a non-degenerate symmetric, or skew symmetric pairing. In addition to these classical cases there are the finite number of exceptional groups. Amazingly enough, motivated by questions in physics, much interest centres around the group \( E_8 \) and its subgroups. For these applications it does not suffice to consider only the classical groups. Thus, while often first doing the case of \( SU(n) \) or more generally of the classical groups, we shall extend our discussions to the general semi-simple group. Also, we shall spend a good deal of time considering elliptically fibered manifolds of the simplest type, namely, elliptic curves.

[75] R. Friedman and J. W. Morgan: “Holomorphic principal bundles over elliptic curves, II: The parabolic construction,” J. Differential Geom. 56 : 2 (2000), pp. 301–379. MR 1863019 Zbl 1033.14016 article

[76] J. de Boer, R. Dijkgraaf, K. Hori, A. Keurentjes, J. Morgan, D. R. Morrison, and S. Sethi: “Triples, fluxes, and strings,” Adv. Theor. Math. Phys. 4 : 5 (2000), pp. 995–1186. MR 1868756 Zbl 1011.81065 article

We study string compactifications with sixteen supersymmetries. The moduli space for these compactifications becomes quite intricate in lower dimensions, partly because there are many different irreducible components. We focus primarily, but not exclusively, on compactifications to seven or more dimensions. These vacua can be realized in a number ways: the perturbative constructions we study include toroidal compactifications of the heterotic/type I strings, asymmetric orbifolds, and orientifolds. In addition, we describe less conventional M and F theory compactifications on smooth spaces. The last class of vacua considered are compactifications on singular spaces with non-trivial discrete fluxes.

We find a number of new components in the string moduli space. Contained in some of these components are M theory compactifications with novel kinds of “frozen” singularities. We are naturally led to conjecture the existence of new dualities relating spaces with different singular geometries and fluxes. As our study of these vacua unfolds, we also learn about additional topics including: F theory on spaces without section, automorphisms of del Pezzo surfaces, and novel physics (and puzzles) from equivariant K-theory. Lastly, we comment on how the data we gain about the M theory three-form might be interpreted.

[77] M. Audin, J. W. Morgan, P. Vogel, and D. Bennequin: Nouveaux invariants en géométrie et en topologie. Edited by F. Dumas, J.-Y. Le Dimet, and S. Paycha. Panoramas et Synthèses [Panoramas and Syntheses] 11. Société Mathématique de France (Paris), 2001. With an afterword by Daniel Bennequin. MR 1882443 Zbl 1007.53066 book

[78] J. W. Morgan: “Seiberg–Witten invariants,” pp. 61–98 in Nouveaux invariants en géométrie et en topologie. Edited by F. Dumas, J.-Y. Le Dimet, and S. Paycha. Panoramas et Synthèses [Panoramas and Syntheses] 11. Société Mathématique de France (Paris), 2001. MR 1882445 Zbl 0994.57028 incollection

Recent years have seen many new invariants of manifolds, invariants defined using moduli spaces of solutions to certain partial differential equations. These include the anti-self-dual equations and Donaldson invariants, the pseudo-holomorphic curve equations of Gromov and the resulting Gromov and Gromov–Witten invariants, quantum cohomology, various Chern–Simon’s invariants, Floer homology, and, the subject of this article, the Seiberg–Witten invariants. We cover the background necessary to define these invariants, give the definition, and then several applications to the topology of smooth four-manifolds including to symplectic four-manifolds and complex algebraic surfaces.

[79] R. Friedman and J. W. Morgan: “On the converse to a theorem of Atiyah and Bott,” J. Algebraic Geom. 11 : 2 (2002), pp. 257–292. MR 1874115 Zbl 1061.14028 article

[80] A. Borel, R. Friedman, and J. W. Morgan: Almost commuting elements in compact Lie groups. Mem. Amer. Math. Soc. 747. American Mathematical Society (Providence, RI), 2002. MR 1895253 Zbl 0993.22002 book

[81] R. Friedman and J. W. Morgan: “Exceptional groups and del Pezzo surfaces,” pp. 101–116 in Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000). Edited by A. Bertram, J. A. Carlson, and H. Kley. Contemp. Math. 312. American Mathematical Society (Providence, RI), 2002. MR 1941576 Zbl 1080.14533 incollection

[82] R. Friedman and J. W. Morgan: “Minuscule representations, invariant polynomials, and spectral covers,” pp. 1–41 in Vector bundles and representation theory (Columbia, MO, 2002). Edited by S. D. Cutkosky, D. Edidin, Z. Qin, and Q. Zhang. Contemp. Math. 322. American Mathematical Society (Providence, RI), 2003. MR 1987737 Zbl 1080.14514 incollection

[83] R. Friedman and J. W. Morgan: “Automorphism sheaves, spectral covers, and the Kostant and Steinberg sections,” pp. 217–244 in Vector bundles and representation theory (Columbia, MO, 2002). Edited by S. D. Cutkosky, D. Edidin, Z. Qin, and Q. Zhang. Contemp. Math. 322. American Mathematical Society (Providence, RI), 2003. MR 1987749 Zbl 1080.14526 incollection

[84] J. W. Morgan: “Definition of the Seiberg–Witten (SW) invariants of 4-manifolds,” pp. 1–11 in Low dimensional topology. Edited by B. Li, S. Wang, and X. Zhao. New Stud. Adv. Math. 3. International Press (Somerville, MA), 2003. MR 2052242 Zbl 1044.57012 incollection

[85] J. W. Morgan: “Computation of SW invariants for certain 4-manifolds,” pp. 13–23 in Low dimensional topology. Edited by B. Li, S. Wang, and X. Zhao. New Stud. Adv. Math. 3. International Press (Somerville, MA), 2003. MR 2052243 Zbl 1044.57013 incollection

[86] J. W. Morgan: “Recent progress on the Poincaré conjecture and the classification of 3-manifolds,” Bull. Amer. Math. Soc. (N.S.) 42 : 1 (2005), pp. 57–78. MR 2115067 Zbl 1100.57016 article

[87] A. Clingher and J. W. Morgan: “Mathematics underlying the F-theory/heterotic string duality in eight dimensions,” Comm. Math. Phys. 254 : 3 (2005), pp. 513–563. MR 2126482 Zbl 1066.14051 article

We give an analytic description of the moduli space of classical vacua for F-theory compactified on elliptic \( K3 \) surfaces, on open tubular regions near the two Type II boundary divisors. The structure of these open sets is related to the total spaces of certain holomorphic theta fibrations over the corresponding boundary divisors. As the two Type II divisors can be naturally identified as moduli spaces of elliptic curves and flat \( G \)-bundles with \[ G = (E_8 \times E_8) \rtimes \mathbb{Z}_2 \quad\text{or}\quad \operatorname{Spin}(32)/\mathbb{Z}_2 ,\] one is led to an analytic isomorphism between these open domains and regions of the moduli spaces of heterotic string theory compactified on the two-torus corresponding to large volumes of the torus. This provides a proof for the classical version of F-theory/Heterotic String Duality in eight dimensions. A description of the Type II boundary points in terms of elliptic stable \( K3 \) surfaces is also given.

[88] J. W. Morgan: “Introduction to supermanifolds,” pp. 163–181 in Quantum field theory, supersymmetry, and enumerative geometry. Edited by D. S. Freed, D. R. Morrison, and I. Singer. IAS/Park City Math. Ser. 11. American Mathematical Society (Providence, RI), 2006. MR 2276909 Zbl 0051.43303 incollection

[89] C. F. Doran and J. W. Morgan: “Mirror symmetry and integral variations of Hodge structure underlying one-parameter families of Calabi–Yau threefolds,” pp. 517–537 in Mirror symmetry V. Edited by N. Yui, S.-T. Yau, and J. D. Lewis. AMS/IP Stud. Adv. Math. 38. American Mathematical Society (Providence, RI), 2006. MR 2282973 Zbl 1116.14005 incollection

[90] C. F. Doran and J. W. Morgan: “Algebraic topology of Calabi–Yau threefolds in toric varieties,” Geom. Topol. 11 (2007), pp. 597–642. MR 2302498 Zbl 1137.14028 article

[91] J. W. Morgan: “The Poincaré conjecture,” pp. 713–736 in International Congress of Mathematicians (Madrid, 2006), vol. I: Plenary lectures and ceremonies. Edited by M. Sanz-Solé, J. Soria, J. L. Varona, and J. Verdera. European Mathematical Society (Zürich), 2007. MR 2334208 Zbl 1154.57014 incollection

[92] J. Morgan and G. Tian: Ricci flow and the Poincaré conjecture, vol. 3. Clay Mathematics Monographs. American Mathematical Society; Clay Mathematics Institute (Providence, RI; Cambridge, MA), 2007. MR 2334563 Zbl 1179.57045 book

[93] J. W. Morgan: “Ricci flow and Thurston’s geometrization conjecture,” pp. 105–137 in Low dimensional topology, vol. 15. Edited by T. S. Mrowka and P. S. Ozsváth. IAS/Park City Math. Ser. American Mathematical Society (Providence, RI), 2009. With notes by Max Lipyanskiy. MR 2503494 Zbl 1195.57036 incollection

Low-dimensional topology has long been a fertile area for the interaction of many different disciplines of mathematics, including differential geometry, hyperbolic geometry, combinatorics, representation theory, global analysis, classical mechanics, and theoretical physics. The Park City Mathematics Institute summer school in 2006 explored in depth the most exciting recent aspects of this interaction, aimed at a broad audience of both graduate students and researchers.

The present volume is based on lectures presented at the summer school on low-dimensional topology. These notes give fresh, concise, and high-level introductions to these developments, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field of low-dimensional topology and to senior researchers wishing to keep up with current developments. The volume begins with notes based on a special lecture by John Milnor about the history of the topology of manifolds. It also contains notes from lectures by Cameron Gordon on the basics of three-manifold topology and surgery problems, Mikhail Khovanov on his homological invariants for knots, John Etnyre on contact geometry, Ron Fintushel and Ron Stern on constructions of exotic four-manifolds, David Gabai on the hyperbolic geometry and the ending lamination theorem, Zoltán Szabó on Heegaard Floer homology for knots and three manifolds, and John Morgan on Hamilton’s and Perelman’s work on Ricci flow and geometrization.

[94] J. W. Morgan and F. T.-H. Fong: Ricci flow and geometrization of 3-manifolds, vol. 53. University Lecture Series. American Mathematical Society (Providence, RI), 2010. MR 2597148 Zbl 1196.53003 book

This book is based on lectures given at Stanford University in 2009. The purpose of the lectures and of the book is to give an introductory overview of how to use Ricci flow and Ricci flow with surgery to establish the Poincaré Conjecture and the more general Geometrization Conjecture for 3-dimensional manifolds. Most of the material is geometric and analytic in nature; a crucial ingredient is understanding singularity development for 3-dimensional Ricci flows and for 3-dimensional Ricci flows with surgery. This understanding is crucial for extending Ricci flows with surgery so that they are defined for all positive time. Once this result is in place, one must study the nature of the time-slices as the time goes to infinity in order to deduce the topological consequences.

[95] P. Griffiths and J. Morgan: Rational homotopy theory and differential forms, 2nd edition. Progress in Mathematics 16. Springer (New York), 2013. Revised second edition of 1981 original. MR 3136262 Zbl 1281.55002 book

[96] J. Morgan and G. Tian: The geometrization conjecture, vol. 5. Clay Mathematics Monographs. American Mathematical Society; Clay Mathematics Institute (Providence, RI; Cambridge, MA), 2014. MR 3186136 Zbl 1302.53001 book

This book gives a complete proof of the geometrization conjecture, which describes all compact 3-manifolds in terms of geometric pieces, i.e., 3-manifolds with locally homogeneous metrics of finite volume. The method is to understand the limits as time goes to infinity of Ricci flow with surgery. The first half of the book is devoted to showing that these limits divide naturally along incompressible tori into pieces on which the metric is converging smoothly to hyperbolic metrics and pieces that are locally more and more volume collapsed. The second half of the book is devoted to showing that the latter pieces are themselves geometric. This is established by showing that the Gromov–Hausdorff limits of sequences of more and more locally volume collapsed 3-manifolds are Alexandrov spaces of dimension at most 2 and then classifying these Alexandrov spaces.

[97] J. W. Morgan: “100 years of topology: Work stimulated by Poincaré’s approach to classifying manifolds,” pp. 7–29 in The Poincaré conjecture. Edited by J. Carlson. Clay Math. Proc. 19. American Mathematical Society (Providence, RI), 2014. MR 3308756 Zbl 1304.55001 incollection

Since its formulation in 1904, the Poincaré Conjecture has stood as a signal problem in topology. As such, it has attracted the attention of the leading topologists of each generation. As I will explain in this lecture, while the purely topological methods used to attack this question did not succeed, they have proved extremely fruitful in resolving closely related questions about manifolds. The Poincaré Conjecture continued to stand unresolved but the progress it generated made topology one of the most exciting and vibrant subjects during the twentieth century. The final irony of this story is that the method of solution comes not from the purely topological approach that Poincaré originally suggested but rather from more geometric and analytic approaches that have their foundations in other aspects of Poincaré’s work. While it is impossible to know for sure what Poincaré would have thought of the history of his conjecture and the nature of the solution, it is natural and pleasing to speculate that he would have completely approved of the method.

This presentation is different from the others in this conference, which will be concerned either with details of the proof of the Poincaré Conjecture or the closely related Geometrization Conjecture or an exposition of related subjects. By and large those presentations will cover more geometric and analytic topics. My presentation mostly covers purely topological material. My aim is to show the background of Poincaré’s work leading up to his conjecture as he grappled with how to understand the topology of manifolds. Then I will explain his direct approach to his conjecture about the 3-sphere and why the direct approach has been so tantalizing to generation after generation of topologists. I will then discuss how the study on manifolds evolved since Poincaré’s time, and what successes successive generations of topologists did have with techniques that can be traced back to Poincaré. Lastly, I will sketch the modern developments where ideas from physics, geometry, and analysis have been brought to bear on the difficult questions about 3- and 4-dimensional manifolds.

[98] G. Brumfiel, A. Medina-Mardones, and J. Morgan: “A cochain level proof of Adem relations in the \( \mathrm{ mod}\,2 \) Steenrod algebra,” J. Homotopy Relat. Struct. 16 : 4 (2021), pp. 517–562. MR 4343073 Zbl 1487.55027 article

In 1947, N. E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-\( i \) products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes \( p \). Steenrod’s student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod 2 Adem relations at the cochain level. Specifically, given a mod 2 cocycle, we produce explicit cochain formulae whose coboundaries are the Adem relations among compositions of Steenrod Squares applied to the cocycle, using Steenrod’s original cochain definition of the Square operations.

John Willard Morgan

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