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Turaev conjectured that the classification, realization and splitting results for Poincaré duality complexes of dimension (–complexes) generalize to –complexes with –connected universal cover for . Baues and Bleile showed that such complexes are classified, up to oriented homotopy equivalence, by the triple consisting of their fundamental group, orientation class and the image of their fundamental class in the homology of the fundamental group, verifying Turaev’s conjecture on classification.
We prove Turaev’s conjectures on realization and splitting. We show that a triple , comprising a group , a cohomology class and a homology class , can be realized by a –complex with –connected universal cover if and only if the Turaev map applied to yields an equivalence. We show that a –complex with –connected universal cover is a nontrivial connected sum of two such complexes if and only if its fundamental group is a nontrivial free product of groups.
We then consider the indecomposable –complexes of this type. When is odd the results are similar to those for the case . The indecomposables are either aspherical or have virtually free fundamental group. When is even the indecomposables include manifolds which are neither aspherical nor have virtually free fundamental group, but if the group is virtually free and has no dihedral subgroup of order then it has two ends.
We prove the following generalization of a classical result of Adams: for any pointed path-connected topological space , that is not necessarily simply connected, the cobar construction of the differential graded (dg) coalgebra of normalized singular chains in with vertices at is weakly equivalent as a differential graded associative algebra (dga) to the singular chains on the Moore based loop space of at . We deduce this statement from several more general categorical results of independent interest. We construct a functor from simplicial sets to categories enriched over cubical sets with connections, which, after triangulation of their mapping spaces, coincides with Lurie’s rigidification functor from simplicial sets to simplicial categories. Taking normalized chains of the mapping spaces of yields a functor from simplicial sets to dg categories which is the left adjoint to the dg nerve functor. For any simplicial set with , is a dga isomorphic to , the cobar construction on the dg coalgebra of normalized chains on . We use these facts to show that sends categorical equivalences between simplicial sets to maps of connected dg coalgebras which induce quasi-isomorphisms of dgas under the cobar functor, which is strictly stronger than saying the resulting dg coalgebras are quasi-isomorphic.
For a finite real reflection group we use noncrossing partitions of type to construct finite cell complexes with the homotopy type of the Milnor fiber of the associated –discriminant and that of the Milnor fiber of the defining polynomial of the associated reflection arrangement. These complexes support natural cyclic group actions realizing the geometric monodromy. Using the shellability of the noncrossing partition lattice, this cell complex yields a chain complex of homology groups computing the integral homology of the Milnor fiber of .
The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex to a singular resolution of a knot . Manolescu conjectured that when is in braid position, the homology is isomorphic to the HOMFLY-PT homology of . Together with a naturality condition on the induced edge maps, this conjecture would prove the existence of a spectral sequence from HOMFLY-PT homology to knot Floer homology. Using a basepoint filtration on , a recursion formula for HOMFLY-PT homology and additional –like differentials on , we prove Manolescu’s conjecture. The naturality condition remains open.
We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the –primary part of is . This was the last unsolved –extension problem left by the recent work of Isaksen and the authors through the –stem.
The proof of this result uses the technique, which was introduced by the authors to prove . This paper advertises this technique through examples that have simpler proofs than in our previous work.
Given a spherical fibration over the classifying space of a finite group we define a dimension function for the –fold fiber join of , where is some large positive integer. We show that the dimension functions satisfy the Borel–Smith conditions when is large enough. As an application we prove that there exists no spherical fibration over the classifying space of with –effective Euler class, generalizing a result of Ünlü (2004) about group actions on finite complexes homotopy equivalent to a sphere. We have been informed that this result will also appear in upcoming work of Alejandro Adem and Jesper Grodal as a corollary of a previously announced program on homotopy group actions due to Grodal.
We expand the theory of weighted sheaves over posets, and use it to study the local homology of Artin groups. First, we use such theory to relate the homology of classical braid groups with the homology of certain independence complexes of graphs. Then, in the context of discrete Morse theory on weighted sheaves, we introduce a particular class of acyclic matchings. Explicit formulas for the homology of the corresponding Morse complexes are given, in terms of the ranks of the associated incidence matrices. We use such method to perform explicit computations for the new affine case , as well as for the cases , and (which were already done before by different methods).
We study the fixed points of the universal –equivariant complex vector bundle of rank and obtain a decomposition formula in terms of twisted equivariant universal complex vector bundles of smaller rank. We use this decomposition to describe the fixed points of the complex equivariant K–theory spectrum and the equivariant unitary bordism groups for adjacent families of subgroups.
We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex groups. To do this, we characterize which JSJ trees of a group in this class admit a cocompact group action with quotient a tree. The conditions are stated in terms of two graphs we associate to the degree refinement of a group in this class. We prove there is a group in this class which is quasi-isometric to a Coxeter group but is not abstractly commensurable to a group generated by finite-order elements. Consequently, the subclass of groups in this class generated by finite-order elements is not quasi-isometrically rigid. We provide necessary conditions for two groups in this class to be abstractly commensurable. We use these conditions to prove there are infinitely many abstract commensurability classes within each quasi-isometry class of this class that contains a group generated by finite-order elements.
Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of , and can be computed using an element of . If we denote by the pullback of the universal cover of , then by a theorem of Deligne, every finite index subgroup of contains . As a consequence, a class in the second cohomology of any finite quotient of can at most enable us to compute the signature of a surface bundle modulo . We show that this is in fact possible and investigate the smallest quotient of that contains this information. This quotient is a nonsplit extension of by an elementary abelian group of order . There is a central extension , and appears as a quotient of the metaplectic double cover . It is an extension of by an almost extraspecial group of order , and has a faithful irreducible complex representation of dimension . Provided , the extension is the universal central extension of . Putting all this together, in Section 4 we provide a recipe for computing the signature modulo , and indicate some consequences.
We prove that every finitely generated group of homeomorphisms of the –dimensional sphere all of whose elements have a finite order which is a power of and is such that there exists a uniform bound for the orders of the group elements is finite. We prove a similar result for groups of area-preserving homeomorphisms without the hypothesis that the orders of group elements are powers of provided there is an element of even order.
Let be an orientable, connected topological surface of infinite type (that is, with infinitely generated fundamental group). The main theorem states that if the genus of is finite and at least , then the isomorphism type of the pure mapping class group associated to , denoted by , detects the homeomorphism type of . As a corollary, every automorphism of is induced by a homeomorphism, which extends a theorem of Ivanov from the finite-type setting. In the process of proving these results, we show that is residually finite if and only if has finite genus, demonstrating that the algebraic structure of can distinguish finite- and infinite-genus surfaces. As an independent result, we also show that fails to be residually finite for any infinite-type surface . In addition, we give a topological generating set for equipped with the compact-open topology. In particular, if has at most one end accumulated by genus, then is topologically generated by Dehn twists, otherwise it is topologically generated by Dehn twists along with handle shifts.
We show that the integral equivariant cohomology Chern numbers completely determine the equivariant geometric unitary bordism classes of closed unitary –manifolds, which gives an affirmative answer to a conjecture posed by Guillemin, Ginzburg and Karshon (Moment maps, cobordisms, and Hamiltonian group actions, Remark H.5 in Appendix H.3), where is a torus. As a further application, we also obtain a satisfactory solution of their Question (A) (Appendix H.1.1) on unitary Hamiltonian –manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber, Panov and Ray and the Kronecker pairing of bordism and cobordism. Our approach heavily exploits Quillen’s geometric interpretation of homotopic unitary cobordism theory. Moreover, this method can also be applied to the study of –equivariant unoriented bordism and can still derive the classical result of tom Dieck.
We show that certain left-orderable groups admit no isolated left orders. The groups we consider are cyclic amalgamations of a free group with a general left-orderable group, the HNN extensions of free groups over cyclic subgroups, and a particular class of one-relator groups. In order to prove the results about orders, we develop perturbation techniques for actions of these groups on the line.
It is known that the quantum invariant of a closed –manifold at is of polynomial order as . Recently, Chen and Yang conjectured that the quantum invariant of a closed hyperbolic –manifold at is of order , where is a normalized complex volume of . We can regard this conjecture as a kind of “volume conjecture”, which is an important topic from the viewpoint that it relates quantum topology and hyperbolic geometry.
In this paper, we give a concrete presentation of the asymptotic expansion of the quantum invariant at for closed hyperbolic –manifolds obtained from the –sphere by integral surgery along the figure-eight knot. In particular, the leading term of the expansion is , which gives a proof of the Chen–Yang conjecture for such –manifolds. Further, the semiclassical part of the expansion is a constant multiple of the square root of the Reidemeister torsion for such –manifolds. We expect that the higher-order coefficients of the expansion would be “new” invariants, which are related to “quantization” of the hyperbolic structure of a closed hyperbolic –manifold.
We relate open book decompositions of a –manifold with its Engel structures. Our main result is, given an open book decomposition of whose binding is a collection of –tori and whose monodromy preserves a framing of a page, the construction of an Engel structure whose isotropic foliation is transverse to the interior of the pages and tangent to the binding.
In particular, the pages are contact manifolds and the monodromy is a compactly supported contactomorphism. As a consequence, on a parallelizable closed –manifold, every open book with toric binding carries in the previous sense an Engel structure. Moreover, we show that among the supported Engel structures we construct, there are loose Engel structures.
For primes we prove an approximation to Cohen, Moore and Neisendorfer’s conjecture that the loops on an Anick space retracts off the double loops on a mod- Moore space. The approximation is then used to answer a question posed by Kitchloo regarding the topology of Kac–Moody groups. We show that, for certain rank- Kac–Moody groups , the based loops on is –locally homotopy equivalent to the product of the loops on a –sphere and the loops on an Anick space.
We construct a Hennings-type logarithmic invariant for restricted quantum at a root of unity. This quantum group is not quasitriangular and hence not ribbon, but factorizable. The invariant is defined for a pair: a –manifold and a colored link inside . The link is split into two parts colored by central elements and by trace classes, or elements in the Hochschild homology of , respectively. The two main ingredients of our construction are the universal invariant of a string link with values in tensor powers of , and the modified trace introduced by the third author with his collaborators and computed on tensor powers of the regular representation. Our invariant is a colored extension of the logarithmic invariant constructed by Jun Murakami.
We give a sufficient condition on the hyperplanes used in the Belolipetsky–Thomson inbreeding construction to obtain nonarithmetic manifolds. We explicitly construct infinitely many examples of such manifolds that are pairwise noncommensurable and estimate their volume.
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