David Gabai
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
David Gabai has a total of 12 co-authors. Here are the top co-authors by number of shared publications:
- Tobias Holck Colding
Dept. of Math., MIT, Cambridge, USA
4 shared publications - Daniel Ketover2 shared publications
- Ryan Budney
Mathematics and Statistics University of Victoria PO BOX 3045
2 shared publications - Andrew Yarmola
Department of Mathematics, Yale University, P. O. Box 208283, 06520-8283 New Haven, CT, USA
1 shared publication - Daniel Hartman
Max Planck Institute for Mathematics in Bonn
1 shared publication - Daniel Ketover
Rutgers University, Busch Campus - Hill Center, Piscataway, USA
1 shared publication
Latest Publications
- Top Cited PapersOpen Access
- 3 March 2026
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Pi
- Vol. 14, e9
Pseudo-isotopies of simply connected 4-manifolds
Perron and Quinn gave independent proofs in 1986 that every topological pseudo-isotopy of a simply-connected, compact topological 4-manifold is isotopic to the identity. Another result of Quinn is that every smooth pseudo-isotopy of a simply-connected, compact, smooth 4-manifold is smoothly stably isotopic to the identity. From this he deduced that $\pi _4(\operatorname {\mathrm {TOP}}(4)/\operatorname {\mathrm {O}}(4)) = 0$ . A replacement criterion is used at a key juncture in Quinn’s proofs, but the justification given for it is incorrect. We provide different arguments that bypass the replacement criterion, thus completing Quinn’s proofs of both the topological and the stable smooth pseudo-isotopy theorems. We discuss the replacement criterion and state it as an open problem.- 3 March 2026
- Published by Cambridge University Press (CUP) in Forum of Mathematics, Pi
- Vol. 14, e9
Top Cited PapersOpen AccessPseudo-isotopies of simply connected 4-manifolds
Perron and Quinn gave independent proofs in 1986 that every topological pseudo-isotopy of a simply-connected, compact topological 4-manifold is isotopic to the identity. Another result of Quinn is that every smooth pseudo-isotopy of a simply-connected, compact, smooth 4-manifold is smoothly stably isotopic to the identity. From this he deduced that $\pi _4(\operatorname {\mathrm {TOP}}(4)/\operatorname {\mathrm {O}}(4)) = 0$ . A replacement criterion is used at a key juncture in Quinn’s proofs, but the justification given for it is incorrect. We provide different arguments that bypass the replacement criterion, thus completing Quinn’s proofs of both the topological and the stable smooth pseudo-isotopy theorems. We discuss the replacement criterion and state it as an open problem. - Open Access
On the automorphism groups of hyperbolic manifolds
Let Diff(N) and Homeo(N) denote the smooth and topological group of automorphisms respectively that fix the boundary of the n-manifold N, pointwise. We show that the (n-4)-th homotopy group of Homeo(S^1 \times D^{n-1}) is not finitely-generated for n >= 4 and in particular the topological mapping-class group of S^1\times D^3 is infinitely generated. We apply this to show that the smooth and topological automorphism groups of finite-volume hyperbolic n-manifolds (when n >= 4) do not have the homotopy-type of finite CW-complexes, results previously known for n >= 11 by Farrell and Jones. In particular, we show that if N is a closed hyperbolic n-manifold, and if Diff_0(N) represents the subgroup of diffeomorphisms that are homotopic to the identity, then the (n-4)-th homotopy group of Diff_0(N) is infinitely generated and hence if n=4, then π_0\Diff_0(N) is infinitely generated with similar results holding topologically.
Open AccessOn the automorphism groups of hyperbolic manifolds
Let Diff(N) and Homeo(N) denote the smooth and topological group of automorphisms respectively that fix the boundary of the n-manifold N, pointwise. We show that the (n-4)-th homotopy group of Homeo(S^1 \times D^{n-1}) is not finitely-generated for n >= 4 and in particular the topological mapping-class group of S^1\times D^3 is infinitely generated. We apply this to show that the smooth and topological automorphism groups of finite-volume hyperbolic n-manifolds (when n >= 4) do not have the homotopy-type of finite CW-complexes, results previously known for n >= 11 by Farrell and Jones. In particular, we show that if N is a closed hyperbolic n-manifold, and if Diff_0(N) represents the subgroup of diffeomorphisms that are homotopic to the identity, then the (n-4)-th homotopy group of Diff_0(N) is infinitely generated and hence if n=4, then π_0\Diff_0(N) is infinitely generated with similar results holding topologically.
- Open Access3-Spheres in the 4-Sphere and Pseudo-Isotopies of $S^1\times S^3$
We offer an approach to the smooth 4-dimensional Schoenflies conjecture via pseudo-isotopy theory.
Open Access3-Spheres in the 4-Sphere and Pseudo-Isotopies of $S^1\times S^3$We offer an approach to the smooth 4-dimensional Schoenflies conjecture via pseudo-isotopy theory.
- 1 February 2022
- Published by Springer Nature in Graphs and Combinatorics
- Vol. 38 (2) , 1-11
The Two-Eyes Lemma: A Linking Problem for Table-Top Necklaces
- 1 February 2022
- Published by Springer Nature in Graphs and Combinatorics
- Vol. 38 (2) , 1-11
The Two-Eyes Lemma: A Linking Problem for Table-Top Necklaces
- Open Access
Knotted 3-balls in S^4
The unknot U in S^4 has non-unique smooth spanning 3-balls up to isotopy fixing U. Equivalently there are properly embedded non-separating 3-balls in S^1xB^3 not properly isotopic to 1xB^3. More generally there exist non-separating 3-spheres in S^1xS^3 not isotopic to 1xS^3 and non trivial elements of \pi_0 Diff_0(S^1xS^3). Along the way we introduce barbell diffeomorphisms, implantations and twistings to construct and modify diffeomorphisms homotopic to the identity. We also introduce a 2-parameter calculus of embeddings of the interval into 4-manifolds and introduce a framed cobordism method as well as a direct method for showing that certain 2-parameter families are homotopically non trivial and diffeomorphisms are isotopically nontrivial. Extensions to higher dimensional manifolds are obtained.
Open AccessKnotted 3-balls in S^4
The unknot U in S^4 has non-unique smooth spanning 3-balls up to isotopy fixing U. Equivalently there are properly embedded non-separating 3-balls in S^1xB^3 not properly isotopic to 1xB^3. More generally there exist non-separating 3-spheres in S^1xS^3 not isotopic to 1xS^3 and non trivial elements of \pi_0 Diff_0(S^1xS^3). Along the way we introduce barbell diffeomorphisms, implantations and twistings to construct and modify diffeomorphisms homotopic to the identity. We also introduce a 2-parameter calculus of embeddings of the interval into 4-manifolds and introduce a framed cobordism method as well as a direct method for showing that certain 2-parameter families are homotopically non trivial and diffeomorphisms are isotopically nontrivial. Extensions to higher dimensional manifolds are obtained.
- Open Access
Self-Referential Discs and the Light Bulb Lemma
We show how self-referential discs in 4-manifolds lead to the construction of pairs of discs with a common geometrically dual sphere which are homotopic rel $\partial$, concordant and coincide near their boundaries, yet are not properly isotopic. This occurs in manifolds without 2-torsion in their fundamental group, e.g. the boundary connect sum of $S^2\times D^2$ and $S^1\times B^3$, thereby exhibiting phenomena not seen with spheres. On the other hand we show that two such discs are isotopic rel $\partial$ if the manifold is simply connected. We construct in $S^2\times D^2\natural S^1\times B^3$ a properly embedded 3-ball properly homotopic to a $z_0\times B^3$ but not properly isotopic to $z_0\times B^3$.Open AccessSelf-Referential Discs and the Light Bulb Lemma
We show how self-referential discs in 4-manifolds lead to the construction of pairs of discs with a common geometrically dual sphere which are homotopic rel $\partial$, concordant and coincide near their boundaries, yet are not properly isotopic. This occurs in manifolds without 2-torsion in their fundamental group, e.g. the boundary connect sum of $S^2\times D^2$ and $S^1\times B^3$, thereby exhibiting phenomena not seen with spheres. On the other hand we show that two such discs are isotopic rel $\partial$ if the manifold is simply connected. We construct in $S^2\times D^2\natural S^1\times B^3$ a properly embedded 3-ball properly homotopic to a $z_0\times B^3$ but not properly isotopic to $z_0\times B^3$. - Open Access
The 4-Dimensional Light Bulb Theorem
For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $\BZ_2$-torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple closed curve in $S^4$ and $\pi_0(\Diff_0(S^2\times D^2)/\Diff_0(B^4))=1$. In manifolds with $\BZ_2$-torsion, one surface can be put into a normal form relative to the other.Open AccessThe 4-Dimensional Light Bulb Theorem
For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $\BZ_2$-torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple closed curve in $S^4$ and $\pi_0(\Diff_0(S^2\times D^2)/\Diff_0(B^4))=1$. In manifolds with $\BZ_2$-torsion, one surface can be put into a normal form relative to the other. - Top Cited Papers
- 15 June 2020
- Published by American Mathematical Society (AMS) in Journal of the American Mathematical Society
- Vol. 33 (3) , 609-652
The 4-dimensional light bulb theorem
For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no Z 2 \mathbb {Z}_2 -torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple closed curve in S 4 S^4 and π 0 ( D i f f 0 ( S 2 × D 2 ) / D i f f 0 ( B 4 ) ) = 1 \pi _0(Diff_0(S^2\times D^2)/Diff_0(B^4))=1 . In manifolds with Z 2 \mathbb {Z}_2 -torsion, one surface can be put into a normal form relative to the other.
- 15 June 2020
- Published by American Mathematical Society (AMS) in Journal of the American Mathematical Society
- Vol. 33 (3) , 609-652
Top Cited PapersThe 4-dimensional light bulb theorem
For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no Z 2 \mathbb {Z}_2 -torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple closed curve in S 4 S^4 and π 0 ( D i f f 0 ( S 2 × D 2 ) / D i f f 0 ( B 4 ) ) = 1 \pi _0(Diff_0(S^2\times D^2)/Diff_0(B^4))=1 . In manifolds with Z 2 \mathbb {Z}_2 -torsion, one surface can be put into a normal form relative to the other.
- Open Access
Geometric Methods In Heegaard Theory
We survey some recent geometric methods for studying Heegaard splittings of 3-manifolds
Open AccessGeometric Methods In Heegaard Theory
We survey some recent geometric methods for studying Heegaard splittings of 3-manifolds
- 1 January 2020
- Published by International Press of Boston in Surveys in Differential Geometry
- Vol. 25 (1) , 33-70
Geometric methods in Heegaard theory
International Press of Boston - publishers of scholarly mathematical and scientific journals and books
- 1 January 2020
- Published by International Press of Boston in Surveys in Differential Geometry
- Vol. 25 (1) , 33-70
Geometric methods in Heegaard theory
International Press of Boston - publishers of scholarly mathematical and scientific journals and books