Celebratio Mathematica

R. C. Kirby: Smoothing locally flat imbeddings. Ph.D. thesis, University of Chicago, 1965. Advised by S. E. Dyer. A short paper based on this work, with the same title, was published in Bull. Am. Math. Soc. 72:1 (1966). MR 2611548 phdthesis

R. C. Kirby: “Smoothing locally flat imbeddings,” Bull. Amer. Math. Soc. 72 (1966), pp. 147–148. MR 0189053 Zbl 0138.18803

R. C. Kirby: “On the annulus conjecture,” Proc. Amer. Math. Soc. 17 (1966), pp. 178–185. MR 0192481 Zbl 0151.32902

R. C. Kirby: “Smoothing locally flat imbeddings of differentiable manifolds,” Topology 6 (1967), pp. 207–220. MR 0211410 Zbl 0152.22401

R. C. Kirby: “The union of flat \( (n-1) \)-balls is flat in \( R^{n} \),” Bull. Amer. Math. Soc. 74 (1968), pp. 614–617. MR 0225328 Zbl 0157.54603

R. C. Kirby: “On the set of non-locally flat points of a submanifold of codimension one,” Ann. of Math. (2) 88 (1968), pp. 281–290. MR 0236900

R. Kirby: Extension to stable homeomorphisms. Single-page typescript (unpublished), 1968. techreport

R. C. Kirby, L. C. Siebenmann, and C. T. C. Wall: “The annulus conjecture and triangulation,” Notices Am. Math. Soc. 16 (1969), pp. 432. article

Laurent C. Siebenmann	Related
C. T. C. Wall	Related

R. C. Kirby: “Stable homeomorphisms and the annulus conjecture,” Ann. of Math. (2) 89 (1969), pp. 575–582. MR 0242165 Zbl 0176.22004

R. C. Kirby and L. C. Siebenmann: “On the triangulation of manifolds and the Hauptvermutung,” Bull. Amer. Math. Soc. 75 (1969), pp. 742–749. MR 0242166 Zbl 0189.54701

R. Kirby and L. Siebenmann: “For manifolds the Hauptvermutung and the triangulation conjecture are false,” Notices Amer. Math. Soc. 16 : 4 (1969), pp. 695. Abstract 69T-G90. article

R. C. Kirby: “Codimension-two locally flat imbeddings have normal bundles,” pp. 416–423 in Topology of manifolds (Univ. of Georgia, Athens, GA, 1969). Markham (Chicago, IL), 1970. MR 0273625

R. D. Edwards and R. C. Kirby: “Deformations of spaces of imbeddings,” Ann. Math. (2) 93 : 1 (January 1971), pp. 63–88. MR 283802 Zbl 0214.50303 article

In [1969], the second author proved that homeomorphisms of the torus \( T^n \) (the \( n \)-fold product of circles) are stable and used this fact, along with an immersion of \( T^n \) minus a point into \( \mathbb{R}^n \), to prove that the homeomorphism group of \( \mathbb{R}^n \) (with the compact-open topology) is locally contractible. This paper generalizes the techniques of that paper and applies them to arbitrary manifolds. The main theorem of the paper says that if \( U \) is a subset of a manifold \( M \) containing a compact subset \( C \) in its interior, then given any two sufficiently close proper imbeddings \[ h_0,g_0:U\to M \] of \( U \) into \( M \), there is a canonical proper isotopy \[ h_t:U\to M , \quad t\in I ,\] connecting \( h_0 \) to an imbedding \( h_1 \), which agrees with \( g_0 \) on \( C \). Furthermore, the isotopy is fixed on the complement of a compact neighborhood of \( C \) in \( U \).

R. C. Kirby and L. C. Siebenmann: “Some theorems on topological manifolds,” pp. 1–7 in Manifolds–Amsterdam 1970. Lecture Notes in Mathematics 197. Springer (Berlin), 1971. MR 0283807 Zbl 0216.45203

R. C. Kirby: “Some conjectures about four-manifolds,” pp. 79–84 in Actes du Congrès International des Mathématiciens (Nice, 1970), tome 2. Gauthier-Villars (Paris), 1971. MR 0431179 Zbl 0223.57004

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
Martin Scharlemann	Related	Volume

R. C. Kirby and L. C. Siebenmann: “Normal bundles for codimension-2 locally flat imbeddings,” pp. 310–324 in Geometric topology (Park City, UT, 1974). Edited by L. C. Glaser and T. B. Rushing. Lecture Notes in Math. 438. Springer (Berlin), 1975. MR 0400237 Zbl 0331.57004

Laurent C. Siebenmann	Related
T. Benny Rushing	Related
Leslie C. Glaser	Related

R. C. Kirby and L. C. Siebenmann: Foundational essays on topological manifolds, smoothings, and triangulations. Annals of Mathematics Studies 88. Princeton University Press, 1977. With notes by John Milnor and Michael Atiyah. MR 0645390 Zbl 0361.57004 book

Laurent C. Siebenmann	Related
M. F. Atiyah	Related	Volume
John W. Milnor	Related

R. Kirby: “A calculus for framed links in \( S^{3} \),” Invent. Math. 45 : 1 (1978), pp. 35–56. MR 0467753 Zbl 0377.55001

R. Kirby and P. Melvin: “Slice knots and property R,” Invent. Math. 45 : 1 (1978), pp. 57–59. MR 0467754 Zbl 0377.55002

incollection M. Freedman and R. Kirby: “A geometric proof of Rochlin’s theorem,” pp. 85–97 in Algebraic and geometric topology (Stanford University, CA, August 2–21, 1976), part 2. Edited by R. J. Milgram. Proceedings of Symposia in Pure Mathematics XXXII. American Mathematical Society (Providence, RI), 1978. MR 0520525 Zbl 0392.57018

In 1974 Andrew Casson outlined to us a proof of Rochlin’s Theorem on the index of a smooth, closed 4-manifold \( M^4 \). His proof involved the Arf invariant of a certain quadratic form defined on the first homology group of a surface in \( M^4 \) which is dual to the second Stiefel–Whitney class of \( M^4 \). Our proof was derived from Casson’s; it is the same in principle but differs considerably in detail. After this manuscript was written, we discovered that Rochlin had already in 1971 given a short sketch of this proof; it appears in a paper [Rochlin 1972] about real algebraic curves in \( \mathbb{R}P^2 \).

In addition we obtain a “stable” converse to the Kervaire–Milnor nonimbedding theorem [Kervaire and Milnor 1961], and in Section 2, by relaxing some orientability assumptions, we prove a new (but unspectacular) nonimbedding theorem and find an obstruction to approximating unoriented simplicial 3-chains in a 5-manifold by an immersed 3-manifold.

Richard James Milgram	Related
M. H. Freedman	Related	Volume

R. Kirby: “Problems in low dimensional manifold theory,” pp. 273–312 in Algebraic and geometric topology (Stanford Univ., CA, 1976), part 2. Edited by R. J. Milgram. Proc. Sympos. Pure Math. XXXII. Amer. Math. Soc. (Providence, R.I.), 1978. MR 520548 Zbl 0394.57002

S. Akbulut and R. Kirby: “An exotic involution of \( S^{4} \),” Topology 18 : 1 (1979), pp. 75–81. MR 528237 Zbl 0465.57013

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
Martin Scharlemann	Related	Volume

R. C. Kirby and W. B. R. Lickorish: “Prime knots and concordance,” Math. Proc. Cambridge Philos. Soc. 86 : 3 (1979), pp. 437–441. MR 542689 Zbl 0426.57001

S. Akbulut and R. Kirby: “Mazur manifolds,” Michigan Math. J. 26 : 3 (1979), pp. 259–284. MR 544597 Zbl 0443.57011

S. Akbulut and R. Kirby: “Branched covers of surfaces in 4-manifolds,” Math. Ann. 252 : 2 (1979/80), pp. 111–131. MR 593626 Zbl 0421.57002

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

Four-manifold theory (Durham, NH, July 4–10, 1982). Edited by C. Gordon and R. Kirby. Contemporary Mathematics 35. American Mathematical Society (Providence, RI), 1984. MR 780574

R. Kirby: “4-manifold problems,” pp. 513–528 in Four-manifold theory (Durham, NH, 4–10 July 1982). Edited by C. Gordon and R. Kirby. Contemporary Mathematics 35. American Mathematical Society (Providece, RI), 1984. Zbl 0558.57004 incollection

S. Akbulut and R. Kirby: “A potential smooth counterexample in dimension 4 to the Poincaré conjecture, the Schoenflies conjecture, and the Andrews–Curtis conjecture,” Topology 24 : 4 (1985), pp. 375–390. MR 816520 Zbl 0584.57009

J. Harer, A. Kas, and R. Kirby: “Handlebody decompositions of complex surfaces,” Mem. Amer. Math. Soc. 62 : 350 (1986), pp. iv+102. MR 849942

John L. Harer	Related
Arnold Samuel Kas	Related

R. C. Kirby: The topology of 4-manifolds. Lecture Notes in Mathematics 1374. Springer (Berlin), 1989. MR 1001966 Zbl 0668.57001

R. C. Kirby and L. R. Taylor: “A calculation of \( \mathrm{Pin}^+ \) bordism groups,” Comment. Math. Helv. 65 : 3 (1990), pp. 434–447. MR 1069818 Zbl 0719.55004

R. Kirby and P. Melvin: “Evaluations of the 3-manifold invariants of Witten and Reshetikhin–Turaev for \( \mathrm{sl}(2,\mathbf{C}) \),” pp. 101–114 in Geometry of low-dimensional manifolds (Durham, 1989), vol. 2. Edited by S. K. Donaldson and C. B. Thomas. London Math. Soc. Lecture Note Ser. 151. Cambridge Univ. Press, 1990. MR 1171911 Zbl 0756.57006

Charles B. Thomas	Related
Paul Michael Melvin	Related
S. K. Donaldson	Related

R. C. Kirby and L. R. Taylor: “\( \mathit{Pin} \) structures on low-dimensional manifolds,” pp. 177–242 in Geometry of low-dimensional manifolds (Durham, 1989), vol. 2. Edited by S. K. Donaldson and C. B. Thomas. London Math. Soc. Lecture Note Ser. 151. Cambridge Univ. Press, 1990. MR 1171915 Zbl 0754.57020

Laurence Robert Taylor	Related
Charles B. Thomas	Related
S. K. Donaldson	Related

R. Kirby: “Reflections on graduate education,” pp. 119–125 in Paul Halmos: Celebrating 50 years of mathematics. Edited by J. H. Ewing and F. W. Gehring. Springer (New York), 1991. incollection

Paul Richard Halmos	Related
Frederick W. Gehring	Related
John H. Ewing	Related

R. Kirby and P. Melvin: “The 3-manifold invariants of Witten and Reshetikhin–Turaev for \( \mathrm{sl}(2,\mathbf{C}) \),” Invent. Math. 105 : 3 (1991), pp. 473–545. MR 1117149 Zbl 0745.57006

R. Kirby, P. Melvin, and X. Zhang: “Quantum invariants at the sixth root of unity,” Comm. Math. Phys. 151 : 3 (1993), pp. 607–617. MR 1207268 Zbl 0779.57007

Paul Michael Melvin	Related
Xingru Zhang	Related

R. Kirby and P. Melvin: “Dedekind sums, \( \mu \)-invariants and the signature cocycle,” Math. Ann. 299 : 2 (1994), pp. 231–267. MR 1275766 Zbl 0809.11027

R. Kirby: “Akbulut’s corks and \( h \)-cobordisms of smooth, simply connected 4-manifolds,” Turkish J. Math. 20 : 1 (1996), pp. 85–93. MR 1392665 Zbl 0868.57031

R. Kirby and P. Melvin: “Canonical framings for 3-manifolds,” pp. 89–115 in Proceedings of 6th Gökova Geometry-Topology Conference, published as Turkish J. Math. 23 : 1. 1999. MR 1701641 Zbl 0947.57020

R. Kirby and P. Melvin: “The \( E_8 \)-manifold, singular fibers and handlebody decompositions,” pp. 233–258 in Proceedings of the Kirbyfest (Berkeley, CA, 1998). Edited by J. Hass and M. Scharlemann. Geom. Topol. Monogr. 2. Geom. Topol. Publ., Coventry, 1999. MR 1734411 Zbl 0959.57020

Paul Michael Melvin	Related
Joel Hass	Related
Martin Scharlemann	Related	Volume

R. Kirby: “Book review: S. Batterson, ‘Stephen Smale: The mathematician who broke the dimension barrier’,” Notices Am. Math. Soc. 47 : 11 (December 2000), pp. 1306–1307. article

Stephen Smale	Related
Steven Lee Batterson	Related

R. C. Kirby and L. R. Taylor: “A survey of 4-manifolds through the eyes of surgery,” pp. 387–421 in Surveys on surgery theory: Papers dedicated to C. T. C. Wall, vol. 2. Edited by S. Cappell, A. Ranicki, and J. Rosenberg. Ann. of Math. Stud. 149. Princeton Univ. Press, 2001. MR 1818779 Zbl 0974.57012

Laurence Robert Taylor	Related
Sylvain Edward Cappell	Related
C. T. C. Wall	Related
Andrew A. Ranicki	Related	Volume

R. Kirby: “Conversations about mathematics,” Notices Amer. Math. Soc. 49 : 3 (2002), pp. 333–335. MR 1879859 Zbl 1126.00302

R. Kirby: “Fleeced?,” Notices Am. Math. Soc. 51 : 2 (February 2004), pp. 1306–1307. article

D. T. Gay and R. Kirby: “Constructing symplectic forms on 4-manifolds which vanish on circles,” Geom. Topol. 8 (2004), pp. 743–777. MR 2057780 Zbl 1054.57027

R. Kirby and P. Melvin: “Local surgery formulas for quantum invariants and the Arf invariant,” pp. 213–233 in Proceedings of the Casson Fest. Edited by C. Gordon and Y. Rieck. Geom. Topol. Monogr. 7. Geom. Topol. Publ. (Coventry), 2004. MR 2172485 Zbl 1087.57006

Andrew J. Casson	Related
Paul Michael Melvin	Related
Yo'av Rieck	Related
Cameron McAllan Gordon	Related	Volume

D. T. Gay and R. Kirby: “Constructing Lefschetz-type fibrations on four-manifolds,” Geom. Topol. 11 (2007), pp. 2075–2115. MR 2350472 Zbl 1135.57009

R. Kirby: “Boy’s surface,” Notices Am. Math. Soc. 54 : 10 (November 2007), pp. 1306–1307. Zbl 1151.53306 article

R. Kirby: “Robion Kirby,” pp. 62–63 in M. Cook: Mathematicians: An outer view of an inner world. Princeton University Press, 2009. incollection

R. C. Kirby: “Low-dimensional topology and geometry,” Proc. Natl. Acad. Sci. USA 108 : 20 (2011), pp. 8081–8084. MR 2806642

D. T. Gay and R. C. Kirby: “Fiber-connected, indefinite Morse 2-functions on connected \( n \)-manifolds,” Proc. Natl. Acad. Sci. USA 108 : 20 (2011), pp. 8122–8125. MR 2806648

J. Crowley, S. Hezlet, R. Kirby, and D. McClure: “Mathematics journals: What is valued and what may change,” Notices Am. Math. Soc. 58 : 8 (2011), pp. 1127–1130. Report on the workshop held at MSRI, Berkeley, CA, 14–16 February 2011. MR 2856147 article

James M. Crowley	Related
Donald Ernest McClure	Related

R. Kirby: Topological manifolds: Their triangulations and smoothings, November 2012. Online series of twelve video lectures from a University of Edinburgh topology seminar. misc

R. Kirby: Morse 2-functions and trisections of 4-manifolds, 27 November 2012. Online video lecture from a University of Edinburgh topology seminar. misc

D. Gay and R. Kirby: “Reconstructing 4-manifolds from Morse 2-functions,” pp. 103–114 in Proceedings of the Freedman Fest (Santa Barbara, CA, 15–17 April 2011 and Berkeley, CA, 6–10 June 2011). Edited by R. Kirby, V. Krushkal, and Z. Wang. Geometry & Topology Monographs 18. Geometry & Topology Publications (Coventry, UK), 2012. Proceedings of a conference on low-dimensional manifolds and high-dimensional categories and a symposium dedicated to Mike Freedman on the occasion of his 60th birthday. MR 3084235 Zbl 1267.57020 ArXiv 1202.3487 incollection

Zhenghan Wang	Related
V. Slava Krushkal	Related
M. H. Freedman	Related	Volume
David T. Gay	Related

@incollection {key3084235m,
	AUTHOR = {Gay, David and Kirby, Robion},
	TITLE = {Reconstructing 4-manifolds from {M}orse
		 2-functions},
	BOOKTITLE = {Proceedings of the {F}reedman {F}est},
	EDITOR = {Kirby, Rob and Krushkal, Vyacheslav
		 and Wang, Zhenghan},
	SERIES = {Geometry \& Topology Monographs},
	NUMBER = {18},
	PUBLISHER = {Geometry \& Topology Publications},
	ADDRESS = {Coventry, UK},
	YEAR = {2012},
	PAGES = {103--114},
	DOI = {10.2140/gtm.2012.18.103},
	NOTE = {(Santa Barbara, CA, 15--17 April 2011
		 and Berkeley, CA, 6--10 June 2011).
		 Proceedings of a conference on low-dimensional
		 manifolds and high-dimensional categories
		 and a symposium dedicated to Mike Freedman
		 on the occasion of his 60th birthday.
		 ArXiv:1202.3487. MR:3084235. Zbl:1267.57020.},
	ISSN = {1464-8989},
}

R. Kirby, P. Melvin, and P. Teichner: “Cohomotopy sets of 4-manifolds,” pp. 161–190 in Proceedings of the Freedman Fest (Santa Barbara, CA, 15–17 April 2011 and Berkeley, CA, 6–10 June 2011). Edited by R. Kirby, V. Krushkal, and Z. Wang. Geometry & Topology Monographs 18. Geometry & Topology Publications (Coventry, UK), 2012. MR 3084237 Zbl 1273.57012 ArXiv 1203.1608 incollection

Zhenghan Wang	Related
V. Slava Krushkal	Related
M. H. Freedman	Related	Volume
Peter Teichner	Related
Paul Michael Melvin	Related

Proceedings of the Freedman Fest (Santa Barbara, CA, 15–17 April 2011 and Berkeley, CA, 6–10 June 2011). Edited by R. Kirby, V. Krushkal, and Z. Wang. Geometry & Topology Monographs 18. Geometry & Topology Publications (Coventry, UK), 2012. Conference on low-dimensional manifolds and high-dimensional categories and symposium dedicated to Mike Freedman on the occasion of his 60th birthday. MR 3137657 Zbl 1253.00022 book

Zhenghan Wang	Related
V. Slava Krushkal	Related
M. H. Freedman	Related	Volume

D. T. Gay and R. Kirby: “Indefinite Morse 2-functions: Broken fibrations and generalizations,” Geom. Topol. 19 : 5 (2015), pp. 2465–2534. MR 3416108 Zbl 1328.57019 article

D. Gay and R. Kirby: “Trisecting 4-manifolds,” Geom. Topol. 20 : 6 (2016), pp. 3097–3132. MR 3590351 Zbl 1372.57033 article

We show that any smooth, closed, oriented, connected 4-manifold can be trisected into three copies of \( \#^k(\mathbb{S}^1\times \mathbb{B}^3) \), intersecting pairwise in 3-dimensional handlebodies, with triple intersection a closed 2-dimensional surface. Such a trisection is unique up to a natural stabilization operation. This is analogous to the existence, and uniqueness up to stabilization, of Heegaard splittings of 3-manifolds. A trisection of a 4-manifold \( X \) arises from a Morse 2-function \( G:X\to \mathbb{B}^2 \) and the obvious trisection of \( \mathbb{B}^2 \), in much the same way that a Heegaard splitting of a 3-manifold \( Y \) arises from a Morse function \( g:Y\to \mathbb{B}^1 \) and the obvious bisection of \( \mathbb{B}^1 \).

R. Kirby: “Celebratio Mathematica,” Notices Am. Math. Soc. 63 : 11 (2016), pp. 1278–1280. Zbl 1352.68277 article

A. Abrams, D. Gay, and R. Kirby: “Group trisections and smooth 4-manifolds,” Geom. Topol. 22 : 3 (2018), pp. 1537–1545. MR 3780440 ArXiv 1605.06731 article

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies and the closed 4-manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the 4-manifold group. A trisected 4-manifold thus gives a trisected group; here we show that every trisected group uniquely determines a trisected 4-manifold. Together with Gay and Kirby’s existence and uniqueness theorem for 4-manifold trisections, this gives a bijection from group trisections modulo isomorphism and a certain stabilization operation to smooth, closed, connected, oriented 4-manifolds modulo diffeomorphism. As a consequence, smooth 4-manifold topology is, in principle, entirely group-theoretic. For example, the smooth 4-dimensional Poincaré conjecture can be reformulated as a purely group-theoretic statement.

Aaron David Abrams	Related
David T. Gay	Related

R. Kirby: “Trisections of 4-manifolds,” Proc. Natl. Acad. Sci. USA 115 : 43 (2018), pp. 10853–10856. MR 3871786 Zbl 1421.57030 article

R. Kirby and A. Thompson: “A new invariant of 4-manifolds,” Proc. Natl. Acad. Sci. USA 115 : 43 (2018), pp. 10857–10860. MR 3871787 Zbl 1421.57031 article

D. Bisch, D. E. Evans, R. Kirby, and S. Popa: “Memories of Vaughan Jones,” Notices Amer. Math. Soc. 68 : 9 (2021), pp. 1540–1563. MR 4323829 Zbl 1480.01028 article

David E. Evans	Related
Dietmar Herbert Bisch	Related
Vaughan F. R. Jones	Related	Volume

R. Kirby and A. Thompson: “Trisections and link surgeries,” New Zealand J. Math. 52 (2021 [2021–2022]), pp. 145–152. MR 4374440 Zbl 1473.57052 article

Robion C. Kirby

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