Sam Nariman | Stanford University (original) (raw)
Papers by Sam Nariman
arXiv (Cornell University), Jan 7, 2024
arXiv (Cornell University), May 30, 2024
We prove that the groups of orientation-preserving homeomorphisms and diffeomorphisms of R n are ... more We prove that the groups of orientation-preserving homeomorphisms and diffeomorphisms of R n are boundedly acyclic, in all regularities. This is the first full computation of the bounded cohomology of a transformation group that is not compactly supported, and it implies that many characteristic classes of flat R n-and S n-bundles are unbounded. We obtain the same result for the group of homeomorphisms of the disc that restrict to the identity on the boundary, and for the homeomorphism group of the non-compact Cantor set. In the appendix, Alexander Kupers proves a controlled version of the annulus theorem which we use to study the bounded cohomology of the homeomorphism group of the discs.
arXiv (Cornell University), Nov 30, 2015
We show that the group cohomology of the diffeomorphisms of the disk with n punctures has the coh... more We show that the group cohomology of the diffeomorphisms of the disk with n punctures has the cohomology of the braid group of n strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding Br 2g+2 ↪ Mod g,2 to a group homomorphism between diffeomorphism groups.
arXiv (Cornell University), Jun 14, 2017
For an oriented manifold M whose dimension is less than 4, we use the contractibility of certain ... more For an oriented manifold M whose dimension is less than 4, we use the contractibility of certain complexes associated to its submanifolds to cut M into simpler pieces in order to do local to global arguments. In particular, in these dimensions, we give a different proof of a deep theorem of Thurston in foliation theory that says the natural map between classifying spaces BHomeo δ (M) → BHomeo(M) induces a homology isomorphism where Homeo δ (M) denotes the group of homeomorphisms of M made discrete. Our proof shows that in low dimensions, Thurston's theorem can be proved without using foliation theory. Finally, we show that this technique gives a new perspective on the homotopy type of homeomorphism groups in low dimensions. In particular, we give a different proof of Hacher's theorem that the homeomorphism groups of Haken 3-manifolds with boundary are homotopically discrete without using his disjunction techniques.
Forum of Mathematics, Sigma, 2023
In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston's th... more In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston's theorem for foliated bundles to prove new variants of this theorem for PL homeomorphisms and contactormorphisms. These versions answer questions posed by Gelfand-Fuks ([GF73, Section 5]) and Greenberg ([Gre92]) on PL foliations and Rybicki ([Ryb10, Section 11]) on contactomorphisms. The interesting point about the original Thurston's technique compared to the better-known Segal-McDuff's proof of the Mather-Thurston theorem is that it gives a compactly supported c-principle theorem without knowing the relevant local statement on open balls. In the appendix, we show that Thurston's fragmentation implies the non-abelian Poincare duality theorem and its generalization using blob complexes ([MW12, Theorem 7.3.1]).
arXiv (Cornell University), May 28, 2019
Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary... more Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary ∂M of a 3-manifold to a C 0-action on M. Among other results, we show that for a 3-manifold M , the S 1 × S 1 action on the boundary does not extend to a C 0-action of S 1 × S 1 as a discrete group on M , except in the trivial case M ≅ D 2 × S 1 .
Forum of Mathematics, Sigma
In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston’s th... more In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston’s theorem for foliated bundles to prove new variants of this theorem for PL homeomorphisms and contactormorphisms. These versions answer questions posed by Gelfand-Fuks ([GF73, Section 5]) and Greenberg ([Gre92]) on PL foliations and Rybicki ([Ryb10, Section 11]) on contactomorphisms. The interesting point about the original Thurston’s technique compared to the better-known Segal-McDuff’s proof of the Mather-Thurston theorem is that it gives a compactly supported c-principle theorem without knowing the relevant local statement on open balls. In the appendix, we show that Thurston’s fragmentation implies the non-abelian Poincare duality theorem and its generalization using blob complexes ([MW12, Theorem 7.3.1]). To the memory of John Mather.
arXiv (Cornell University), Mar 13, 2023
In his work on the generalization of the Reeb stability theorem ([Thu74]), Thurston conjectured t... more In his work on the generalization of the Reeb stability theorem ([Thu74]), Thurston conjectured that if the fundamental group of a compact leaf L in a codimension-one transversely orientable foliation is amenable and if the first cohomology group H 1 (L; R) is trivial, then L has a neighborhood foliated as a product. This was later proved as a consequence of Witte-Morris' theorem on the local indicability of amenable left orderable groups and Navas' theorem on the left orderability of the group of germs of orientation-preserving homeomorphisms of the real line at the origin. In this note, we prove that Thurston's conjecture also holds for any foliation that is sufficiently close to the original foliation. Hence, if the fundamental group π1(L) is amenable and H 1 (L; R) = 0, then for every transversely orientable codimension-one foliation F having L as a leaf, there is a neighborhood of F in the space of C 1,0 foliations with Epstein C 0 topology consisting entirely of foliations that are locally a product L × R.
arXiv (Cornell University), Nov 8, 2020
In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston's th... more In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston's theorem for foliated bundles to prove new variants of this theorem for PL homeomorphisms and contactormorphisms. These versions answer questions posed by Gelfand-Fuks ([GF73, Section 5]) and Greenberg ([Gre92]) on PL foliations and Rybicki ([Ryb10, Section 11]) on contactomorphisms. The interesting point about the original Thurston's technique compared to the better-known Segal-McDuff's proof of the Mather-Thurston theorem is that it gives a compactly supported c-principle theorem without knowing the relevant local statement on open balls. In the appendix, we show that Thurston's fragmentation implies the non-abelian Poincare duality theorem and its generalization using blob complexes ([MW12, Theorem 7.3.1]).
![Research paper thumbnail of Topological aspects of the dynamical moduli space of rational maps. (arXiv:1908.10792v1 [math.AT])](https://attachments.academia-assets.com/100721372/thumbnails/1.jpg)
](arXiv (Cornell University), Nov 25, 2019
We investigate the topology of the space of Möbius conjugacy classes of degree d rational maps on... more We investigate the topology of the space of Möbius conjugacy classes of degree d rational maps on the Riemann sphere. We show that it is rationally acyclic and we compute its fundamental group. As a byproduct, we also obtain the ranks of some higher homotopy groups of the parameter space of degree d rational maps allowing us to extend the previously known range. Contents 1. Introduction 2. Background on the spaces Rat d and M d 2.1. The parameter space Rat d 2.2. The moduli space M d 3. The homology of M d 3.1. The rational homology of M d 3.2. Application to marked moduli spaces of rational maps 13 3.3. Variants M pre d and M post d of the moduli space M d 15 3.4. The cohomology of M d with finite coefficients 17 4. The fundamental group of M d 22 5. The higher homotopy groups of Rat d 27 References 32
arXiv (Cornell University), Jul 11, 2022
Haefliger-Thurston's conjecture predicts that Haefliger's classifying space for C r-foliations of... more Haefliger-Thurston's conjecture predicts that Haefliger's classifying space for C r-foliations of codimension n whose normal bundles are trivial is 2n-connected. In this paper, we confirm this conjecture for PL foliations of codimension 2. As a consequence, we use a version of Mather-Thurston's theorem for PL homeomorphisms due to the author to derive new homological properties for PL surface homeomorphisms. In particular, we answer a question of Epstein in dimension 2 and prove the simplicity of the identity component of PL surface homeomorphisms.
Answering a question of Hatcher-Vogtmann, we prove that the top homology group of the free factor... more Answering a question of Hatcher-Vogtmann, we prove that the top homology group of the free factor complex is not the dualizing module for Aut(Fn), at least for n = 5.
We investigate low homological consequences of a conjecture due to Haefliger and Thurston in the ... more We investigate low homological consequences of a conjecture due to Haefliger and Thurston in the context of foliated manifold bundles. In particular, Haefliger-Thurston's conjecture predicts that every MMM-bundle over a manifold BBB where textdim(B)leqtextdim(M)\text{dim}(B)\leq \text{dim}(M)textdim(B)leqtextdim(M) is cobordant to a flat MMM-bundle. We prove this prediction "up to torsion" when BBB is a 333-manifold and for all textdim(M)\text{dim}(M)textdim(M) which is not 111 modulo 444. We also consider the case of PL foliations of codimension 222 and Haefliger-Thurston's conjecture in this case says that the classifying space overlinemathrmBΓ2PL\overline{\mathrm{B}Γ}_2^{PL}overlinemathrmBΓ2PL is 444-connected. We show that this classifying space is 333-connected and π4(overlinemathrmBΓ2PL)otimesmathbbFp=0π_4(\overline{\mathrm{B}Γ}_2^{PL})\otimes \mathbb{F}_p=0π4(overlinemathrmBΓ2PL)otimesmathbbFp=0 for all prime ppp. As a consequence, we answer a question of Epstein regarding the simplicity of the identity component of PL homeomorphisms in dimension 222.
Journal of Topology and Analysis, Mar 1, 2018
Apparently a lost theorem of Thurston ([1]) states that the cube of the Euler class e 3 ∈ H 6 (BD... more Apparently a lost theorem of Thurston ([1]) states that the cube of the Euler class e 3 ∈ H 6 (BDiff δ ω (S 1); Q) is zero where Diff δ ω (S 1) is the analytic orientation preserving diffeomorphisms of the circle with the discrete topology. This is in contrast with Morita's theorem ([5]) that the powers of the Euler class are nonzero in H * (BDiff δ (S 1); Q) where Diff δ (S 1) is the orientation preserving C ∞-diffeomorphisms of the circle with the discrete topology. The purpose of this short note is to prove that the powers of the Euler class e k ∈ H * (BDiff δ ω (S 1); Z) in fact are nonzero in cohomology with integer coefficients. We also give a short proof of Morita's theorem ([5]).
arXiv: Algebraic Topology, Nov 30, 2016
In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of ... more In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. Similar to discrete surface diffeomorphisms [Nar17b], we construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology of certain infinite loop spaces. We use these infinite loop spaces to study characteristic classes of surface bundles whose holonomy groups are area preserving, in particular we give a homotopy theoretic proof of the main theorem in [KM07].
arXiv: Geometric Topology, Nov 8, 2021
We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional man... more We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, it is isomorphic to the polynomial ring generated by the bounded Euler class. These seem to be the first examples of groups for which the entire bounded cohomology can be described without being trivial. We further prove that, contrary to ordinary cohomology, the diffeomorphisms groups of the circle and of the closed 2-disc have the same bounded cohomology as their homeomorphism groups and that both differ from the ordinary cohomology. Finally, we determine the low-dimensional bounded cohomology of homeoand diffeomorphism of the spheres S n and of certain 3-manifolds. In particular, we answer a question of Ghys by showing that the Euler class in H 4 (Homeo○(S 3)) is unbounded.
arXiv: Geometric Topology, Apr 26, 2021
In this paper, we use homological techniques and a theorem of Thurston to study homological finit... more In this paper, we use homological techniques and a theorem of Thurston to study homological finiteness of BDiffpM, rel Bq when M is a reducible 3-manifold with a non-empty boundary that has distinct irreducible factors. Kontsevich ([Kir95, Problem 3.48]) conjectured that BDiffpM, rel Bq has the homotopy type of a finite CW complex for all 3-manifolds with non-empty boundary. Hatcher-McCullough ([HM97]) proved this conjecture when M is irreducible. We show that when M is reducible with non-empty boundary that has distinct irreducible factors, BDiffpM, rel Bq is homology isomorphic to a finite CW-complex.
We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional man... more We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2disc, we show that the bounded cohomology is isomorphic to the polynomial ring generated by the bounded Euler class. We further prove that the group of orientation-preserving homeomorphisms of the line is boundedly acyclic. We also calculate the low-dimensional bounded cohomology of Homeo(S), of Homeo(S) and of homeomorphism groups of certain 3-manifolds.
We prove that the group homology of the diffeomorphism group of #gS n × S n int(D 2n) as a discre... more We prove that the group homology of the diffeomorphism group of #gS n × S n int(D 2n) as a discrete group is independent of g in a range, provided that n > 2. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford-Morita-Miller classes can be detected on flat (#gS n × S n)-bundles for g ≫ 0.
Mathematische Annalen, 2020
Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary... more Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary ∂M of a 3-manifold to a C 0-action on M. Among other results, we show that for a 3-manifold M , the S 1 × S 1 action on the boundary does not extend to a C 0-action of S 1 × S 1 as a discrete group on M , except in the trivial case M ≅ D 2 × S 1 .
arXiv (Cornell University), Jan 7, 2024
arXiv (Cornell University), May 30, 2024
We prove that the groups of orientation-preserving homeomorphisms and diffeomorphisms of R n are ... more We prove that the groups of orientation-preserving homeomorphisms and diffeomorphisms of R n are boundedly acyclic, in all regularities. This is the first full computation of the bounded cohomology of a transformation group that is not compactly supported, and it implies that many characteristic classes of flat R n-and S n-bundles are unbounded. We obtain the same result for the group of homeomorphisms of the disc that restrict to the identity on the boundary, and for the homeomorphism group of the non-compact Cantor set. In the appendix, Alexander Kupers proves a controlled version of the annulus theorem which we use to study the bounded cohomology of the homeomorphism group of the discs.
arXiv (Cornell University), Nov 30, 2015
We show that the group cohomology of the diffeomorphisms of the disk with n punctures has the coh... more We show that the group cohomology of the diffeomorphisms of the disk with n punctures has the cohomology of the braid group of n strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding Br 2g+2 ↪ Mod g,2 to a group homomorphism between diffeomorphism groups.
arXiv (Cornell University), Jun 14, 2017
For an oriented manifold M whose dimension is less than 4, we use the contractibility of certain ... more For an oriented manifold M whose dimension is less than 4, we use the contractibility of certain complexes associated to its submanifolds to cut M into simpler pieces in order to do local to global arguments. In particular, in these dimensions, we give a different proof of a deep theorem of Thurston in foliation theory that says the natural map between classifying spaces BHomeo δ (M) → BHomeo(M) induces a homology isomorphism where Homeo δ (M) denotes the group of homeomorphisms of M made discrete. Our proof shows that in low dimensions, Thurston's theorem can be proved without using foliation theory. Finally, we show that this technique gives a new perspective on the homotopy type of homeomorphism groups in low dimensions. In particular, we give a different proof of Hacher's theorem that the homeomorphism groups of Haken 3-manifolds with boundary are homotopically discrete without using his disjunction techniques.
Forum of Mathematics, Sigma, 2023
In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston's th... more In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston's theorem for foliated bundles to prove new variants of this theorem for PL homeomorphisms and contactormorphisms. These versions answer questions posed by Gelfand-Fuks ([GF73, Section 5]) and Greenberg ([Gre92]) on PL foliations and Rybicki ([Ryb10, Section 11]) on contactomorphisms. The interesting point about the original Thurston's technique compared to the better-known Segal-McDuff's proof of the Mather-Thurston theorem is that it gives a compactly supported c-principle theorem without knowing the relevant local statement on open balls. In the appendix, we show that Thurston's fragmentation implies the non-abelian Poincare duality theorem and its generalization using blob complexes ([MW12, Theorem 7.3.1]).
arXiv (Cornell University), May 28, 2019
Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary... more Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary ∂M of a 3-manifold to a C 0-action on M. Among other results, we show that for a 3-manifold M , the S 1 × S 1 action on the boundary does not extend to a C 0-action of S 1 × S 1 as a discrete group on M , except in the trivial case M ≅ D 2 × S 1 .
Forum of Mathematics, Sigma
In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston’s th... more In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston’s theorem for foliated bundles to prove new variants of this theorem for PL homeomorphisms and contactormorphisms. These versions answer questions posed by Gelfand-Fuks ([GF73, Section 5]) and Greenberg ([Gre92]) on PL foliations and Rybicki ([Ryb10, Section 11]) on contactomorphisms. The interesting point about the original Thurston’s technique compared to the better-known Segal-McDuff’s proof of the Mather-Thurston theorem is that it gives a compactly supported c-principle theorem without knowing the relevant local statement on open balls. In the appendix, we show that Thurston’s fragmentation implies the non-abelian Poincare duality theorem and its generalization using blob complexes ([MW12, Theorem 7.3.1]). To the memory of John Mather.
arXiv (Cornell University), Mar 13, 2023
In his work on the generalization of the Reeb stability theorem ([Thu74]), Thurston conjectured t... more In his work on the generalization of the Reeb stability theorem ([Thu74]), Thurston conjectured that if the fundamental group of a compact leaf L in a codimension-one transversely orientable foliation is amenable and if the first cohomology group H 1 (L; R) is trivial, then L has a neighborhood foliated as a product. This was later proved as a consequence of Witte-Morris' theorem on the local indicability of amenable left orderable groups and Navas' theorem on the left orderability of the group of germs of orientation-preserving homeomorphisms of the real line at the origin. In this note, we prove that Thurston's conjecture also holds for any foliation that is sufficiently close to the original foliation. Hence, if the fundamental group π1(L) is amenable and H 1 (L; R) = 0, then for every transversely orientable codimension-one foliation F having L as a leaf, there is a neighborhood of F in the space of C 1,0 foliations with Epstein C 0 topology consisting entirely of foliations that are locally a product L × R.
arXiv (Cornell University), Nov 8, 2020
In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston's th... more In this paper, we generalize the original idea of Thurston for the so-called Mather-Thurston's theorem for foliated bundles to prove new variants of this theorem for PL homeomorphisms and contactormorphisms. These versions answer questions posed by Gelfand-Fuks ([GF73, Section 5]) and Greenberg ([Gre92]) on PL foliations and Rybicki ([Ryb10, Section 11]) on contactomorphisms. The interesting point about the original Thurston's technique compared to the better-known Segal-McDuff's proof of the Mather-Thurston theorem is that it gives a compactly supported c-principle theorem without knowing the relevant local statement on open balls. In the appendix, we show that Thurston's fragmentation implies the non-abelian Poincare duality theorem and its generalization using blob complexes ([MW12, Theorem 7.3.1]).
arXiv (Cornell University), Nov 25, 2019
We investigate the topology of the space of Möbius conjugacy classes of degree d rational maps on... more We investigate the topology of the space of Möbius conjugacy classes of degree d rational maps on the Riemann sphere. We show that it is rationally acyclic and we compute its fundamental group. As a byproduct, we also obtain the ranks of some higher homotopy groups of the parameter space of degree d rational maps allowing us to extend the previously known range. Contents 1. Introduction 2. Background on the spaces Rat d and M d 2.1. The parameter space Rat d 2.2. The moduli space M d 3. The homology of M d 3.1. The rational homology of M d 3.2. Application to marked moduli spaces of rational maps 13 3.3. Variants M pre d and M post d of the moduli space M d 15 3.4. The cohomology of M d with finite coefficients 17 4. The fundamental group of M d 22 5. The higher homotopy groups of Rat d 27 References 32
arXiv (Cornell University), Jul 11, 2022
Haefliger-Thurston's conjecture predicts that Haefliger's classifying space for C r-foliations of... more Haefliger-Thurston's conjecture predicts that Haefliger's classifying space for C r-foliations of codimension n whose normal bundles are trivial is 2n-connected. In this paper, we confirm this conjecture for PL foliations of codimension 2. As a consequence, we use a version of Mather-Thurston's theorem for PL homeomorphisms due to the author to derive new homological properties for PL surface homeomorphisms. In particular, we answer a question of Epstein in dimension 2 and prove the simplicity of the identity component of PL surface homeomorphisms.
Answering a question of Hatcher-Vogtmann, we prove that the top homology group of the free factor... more Answering a question of Hatcher-Vogtmann, we prove that the top homology group of the free factor complex is not the dualizing module for Aut(Fn), at least for n = 5.
We investigate low homological consequences of a conjecture due to Haefliger and Thurston in the ... more We investigate low homological consequences of a conjecture due to Haefliger and Thurston in the context of foliated manifold bundles. In particular, Haefliger-Thurston's conjecture predicts that every MMM-bundle over a manifold BBB where textdim(B)leqtextdim(M)\text{dim}(B)\leq \text{dim}(M)textdim(B)leqtextdim(M) is cobordant to a flat MMM-bundle. We prove this prediction "up to torsion" when BBB is a 333-manifold and for all textdim(M)\text{dim}(M)textdim(M) which is not 111 modulo 444. We also consider the case of PL foliations of codimension 222 and Haefliger-Thurston's conjecture in this case says that the classifying space overlinemathrmBΓ2PL\overline{\mathrm{B}Γ}_2^{PL}overlinemathrmBΓ2PL is 444-connected. We show that this classifying space is 333-connected and π4(overlinemathrmBΓ2PL)otimesmathbbFp=0π_4(\overline{\mathrm{B}Γ}_2^{PL})\otimes \mathbb{F}_p=0π4(overlinemathrmBΓ2PL)otimesmathbbFp=0 for all prime ppp. As a consequence, we answer a question of Epstein regarding the simplicity of the identity component of PL homeomorphisms in dimension 222.
Journal of Topology and Analysis, Mar 1, 2018
Apparently a lost theorem of Thurston ([1]) states that the cube of the Euler class e 3 ∈ H 6 (BD... more Apparently a lost theorem of Thurston ([1]) states that the cube of the Euler class e 3 ∈ H 6 (BDiff δ ω (S 1); Q) is zero where Diff δ ω (S 1) is the analytic orientation preserving diffeomorphisms of the circle with the discrete topology. This is in contrast with Morita's theorem ([5]) that the powers of the Euler class are nonzero in H * (BDiff δ (S 1); Q) where Diff δ (S 1) is the orientation preserving C ∞-diffeomorphisms of the circle with the discrete topology. The purpose of this short note is to prove that the powers of the Euler class e k ∈ H * (BDiff δ ω (S 1); Z) in fact are nonzero in cohomology with integer coefficients. We also give a short proof of Morita's theorem ([5]).
arXiv: Algebraic Topology, Nov 30, 2016
In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of ... more In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. Similar to discrete surface diffeomorphisms [Nar17b], we construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology of certain infinite loop spaces. We use these infinite loop spaces to study characteristic classes of surface bundles whose holonomy groups are area preserving, in particular we give a homotopy theoretic proof of the main theorem in [KM07].
arXiv: Geometric Topology, Nov 8, 2021
We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional man... more We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, it is isomorphic to the polynomial ring generated by the bounded Euler class. These seem to be the first examples of groups for which the entire bounded cohomology can be described without being trivial. We further prove that, contrary to ordinary cohomology, the diffeomorphisms groups of the circle and of the closed 2-disc have the same bounded cohomology as their homeomorphism groups and that both differ from the ordinary cohomology. Finally, we determine the low-dimensional bounded cohomology of homeoand diffeomorphism of the spheres S n and of certain 3-manifolds. In particular, we answer a question of Ghys by showing that the Euler class in H 4 (Homeo○(S 3)) is unbounded.
arXiv: Geometric Topology, Apr 26, 2021
In this paper, we use homological techniques and a theorem of Thurston to study homological finit... more In this paper, we use homological techniques and a theorem of Thurston to study homological finiteness of BDiffpM, rel Bq when M is a reducible 3-manifold with a non-empty boundary that has distinct irreducible factors. Kontsevich ([Kir95, Problem 3.48]) conjectured that BDiffpM, rel Bq has the homotopy type of a finite CW complex for all 3-manifolds with non-empty boundary. Hatcher-McCullough ([HM97]) proved this conjecture when M is irreducible. We show that when M is reducible with non-empty boundary that has distinct irreducible factors, BDiffpM, rel Bq is homology isomorphic to a finite CW-complex.
We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional man... more We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2disc, we show that the bounded cohomology is isomorphic to the polynomial ring generated by the bounded Euler class. We further prove that the group of orientation-preserving homeomorphisms of the line is boundedly acyclic. We also calculate the low-dimensional bounded cohomology of Homeo(S), of Homeo(S) and of homeomorphism groups of certain 3-manifolds.
We prove that the group homology of the diffeomorphism group of #gS n × S n int(D 2n) as a discre... more We prove that the group homology of the diffeomorphism group of #gS n × S n int(D 2n) as a discrete group is independent of g in a range, provided that n > 2. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford-Morita-Miller classes can be detected on flat (#gS n × S n)-bundles for g ≫ 0.
Mathematische Annalen, 2020
Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary... more Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary ∂M of a 3-manifold to a C 0-action on M. Among other results, we show that for a 3-manifold M , the S 1 × S 1 action on the boundary does not extend to a C 0-action of S 1 × S 1 as a discrete group on M , except in the trivial case M ≅ D 2 × S 1 .