Quotient Complex Research Articles

ALVIS–CURTIS DUALITY ON LOWERCASE q-SCHUR AND HECKE ALGEBRAS

We show that for a finite reductive group tensoring with a quotient of the complex inducing the Alvis?Curtis character duality induces a self-derived equivalence on a quotient of the group algebra. In the case of general linear groups, this equivalence realizes an equivalence between derived categories of q-Schur algebras. Furthermore, in the latter case, multiplication of the quotient complex by an idempotent induces an isomorphism with the two-sided q-analogue of the Coxeter complex.

A graded Gersten–Witt complex for schemes with a dualizing complex and the Chow group

We construct for any scheme X with a dualizing complex I • a Gersten–Witt complex GW ( X , I • ) and show that the differential of this complex respects the filtration by the powers of the fundamental ideal. To prove this we introduce second residue maps for one-dimensional local domains which have a dualizing complex. This residue maps generalize the classical second residue morphisms for discrete valuation rings. For the cohomology of the quotient complexes GrGW p ( X , I • ) of this filtration we prove H p ( GrGW p ( X , I • ) ) ≃ CH p ( X , μ I ) / 2 , where μ I is the codimension function of the dualizing complex I • and CH p ( X , μ I ) denotes the Chow group of μ I -codimension p -cycles modulo rational equivalence.

Cut vertices in commutative graphs

The homology of Kontsevich's commutative graph complex parameterizes finite type invariants of odd dimensional manifolds. This {\it graph homology} is also the twisted homology of Outer Space modulo its boundary, so gives a nice point of contact between geometric group theory and quantum topology. In this paper we give two different proofs (one algebraic, one geometric) that the commutative graph complex is quasi-isomorphic to the quotient complex obtained by modding out by graphs with cut vertices. This quotient complex has the advantage of being smaller and hence more practical for computations. In addition, it supports a Lie bialgebra structure coming from a bracket and cobracket we defined in a previous paper. As an application, we compute the rational homology groups of the commutative graph complex up to rank 7.

A partitioning and related properties for the quotient complex [formula omitted

We study the quotient complex Δ(B lm)/S l≀S m as a means of deducing facts about the ring k[x 1,…,x lm] S l≀S m . It is shown in Hersh (preprint, 2000) that Δ(B lm)/S l≀S m is shellable when l=2, implying Cohen–Macaulayness of k[x 1,…,x 2m] S 2≀S m for any field k. We now confirm for all pairs ( l, m) with l>2 and m>1 that Δ(B lm)/S l≀S m is not Cohen–Macaulay over Z/2 Z , but it is Cohen–Macaulay over fields of characteristic p> m (independent of l). This yields corresponding characteristic-dependent results for k[x 1,…,x lm] S l≀S m . We also prove that Δ(B lm)/S l≀S m and the links of many of its faces are collapsible, and we give a partitioning for Δ(B lm)/S l≀S m .

Lexicographic Shellability for Balanced Complexes

We introduce a notion of lexicographic shellability for pure, balanced boolean cell complexes, modelled after the CL-shellability criterion of Bjorner and Wachs (Adv. in Math. 43 (1982), 87–100) for posets and its generalization by Kozlov (Ann. of Comp. 1(1) (1997), 67–90) called CC-shellability. We give a lexicographic shelling for the quotient of the order complex of a Boolean algebra of rank 2n by the action of the wreath product S2 w Sn of symmetric groups, and we provide a partitioning for the quotient complex Δ(Πn)/Sn. Stanley asked for a description of the symmetric group representation βS on the homology of the rank-selected partition lattice ΠnS in Stanley (J. Combin. Theory Ser. A 32(2) (1982), 132–161), and in particular he asked when the multiplicity bS(n) of the trivial representation in βS is 0. One consequence of the partitioning for Δ(Πn)/Sn is a (fairly complicated) combinatorial interpretation for bS(n)s another is a simple proof of Hanlon's result (European J. Combin. 4(2) (1983), 137–141) that b1,…,i(n) e 0. Using a result of Garsia and Stanton from (Adv. in Math. 51(2) (1984), 107–201), we deduce from our shelling for Δ(B2n)/S2 w Sn that the ring of invariants k[x1,…,x2n]S2 w Sn is Cohen-Macaulay over any field k.

Twisted face-pairing 3-manifolds

This paper is an enriched version of our introductory paper on twisted face-pairing 3-manifolds. Just as every edge-pairing of a 2-dimensional disk yields a closed 2-manifold, so also every face-pairing e of a faceted 3-ball P yields a closed 3-dimensional pseudomanifold. In dimension 3, the pseudomanifold may suffer from the defect that it fails to be a true 3-manifold at some of its vertices. The method of twisted face-pairing shows how to correct this defect of the quotient pseudomanifold P/∈ systematically. The method describes how to modify P by edge subdivision and how to modify any orientation-reversing face-pairing ∈ of P by twisting, so as to yield an infinite parametrized family of face-pairings (Q, δ) whose quotient complexes Q/δ are all closed orientable 3-manifolds. The method is so efficient that, starting even with almost trivial face-pairings ∈, it yields a rich family of highly nontrivial, yet relatively simple, 3-manifolds. This paper solves two problems raised by the introductory paper: (1) Replace the computational proof of the introductory paper by a conceptual geometric proof of the fact that the quotient complex Q/δ of a twisted face-pairing is a closed 3-manifold. We do so by showing that the quotient complex has just one vertex and that its link is the faceted sphere dual to Q. (2) The twist construction has an ambiguity which allows one to twist all faces clockwise or to twist all faces counterclockwise. The fundamental groups of the two resulting quotient complexes are not at all obviously isomorphic. Are the two manifolds the same, or are they distinct? We prove the highly nonobvious fact that clockwise twists and counterclockwise twists yield the same manifold. The homeomorphism between them is a duality homeomorphism which reverses orientation and interchanges natural 0-handles with 3-handles, natural 1-handles with 2-handles. This duality result of (2) is central to our further studies of twisted face-pairings. We also relate the fundamental groups and homology groups of the twisted face-pairing 3-manifolds Q/δ and of the original pseudomanifold P/∈ (with vertices removed). We conclude the paper by giving examples of twisted face-pairing 3-manifolds. These examples include manifolds from five of Thurston's eight 3-dimensional geometries.

Quotient Complexes and Lexicographic Shellability

Let Π n,k,k and Π n,k,h , h < k, denote the intersection lattices of the k-equal subspace arrangement of type $$\mathcal{D}$$ n and the k,h-equal subspace arrangement of type $$\mathcal{B}$$ n respectively. Denote by $$S_n^B $$ the group of signed permutations. We show that Δ(Π n,k,k )/ $$S_n^B $$ is collapsible. For Δ(Π n,k,h )/ $$S_n^B $$ , h < k, we show the following. If n ≡ 0 (mod k), then it is homotopy equivalent to a sphere of dimension $$\tfrac{{2n}}{k}$$ . If n ≡ h (mod k), then it is homotopy equivalent to a sphere of dimension $$2\tfrac{{n - h}}{k} - 1$$ . Otherwise, it is contractible. Immediate consequences for the multiplicity of the trivial characters in the representations of $$S_n^B $$ on the homology groups of Δ(Π n,k,k ) and Δ(Π n,k,h ) are stated. The collapsibility of Δ(Π n,k,k )/ $$S_n^B $$ is established using a discrete Morse function. The same method is used to show that Δ(Π n,k,h )/ $$S_n^B $$ , h < k, is homotopy equivalent to a certain subcomplex. The homotopy type of this subcomplex is calculated by showing that it is shellable. To do this, we are led to introduce a lexicographic shelling condition for balanced cell complexes of boolean type. This extends to the non-pure case work of P. Hersh (Preprint, 2001) and specializes to the CL-shellability of A. Björner and M. Wachs (Trans. Amer. Math. Soc. 4 (1996), 1299–1327) when the cell complex is an order complex of a poset.

COMPARISON OF MULTIGRADED AND UNGRADED COUSIN COMPLEXES

AbstractThis paper generalizes, in two senses, work of Petzl and Sharp, who showed that, for a $\mathbb{Z}$-graded module $M$ over a $\mathbb{Z}$-graded commutative Noetherian ring $R$, the graded Cousin complex for $M$ introduced by Goto and Watanabe can be regarded as a subcomplex of the ordinary Cousin complex studied by Sharp, and that the resulting quotient complex is always exact. The generalizations considered in this paper are, firstly, to multigraded situations and, secondly, to Cousin complexes with respect to more general filtrations than the basic ones considered by Petzl and Sharp. New arguments are presented to provide a sufficient condition for the exactness of the quotient complex in this generality, as the arguments of Petzl and Sharp will not work for this situation without additional input.AMS 2000 Mathematics subject classification: Primary 13A02; 13E05; 13D25; 13D45

Complex hyperbolic manifolds homotopy equivalent to a Riemann surface

We construct isometric actions of fundamental groups of closed Riemann surfaces on the complex hyperbolic plane, which realize all possible values of Toledo’s invariant τ . For integer values of τ these actions are discrete embeddings. The quotient complex hyperbolic surfaces are disc bundles over closed Riemann surfaces, whose topological type is described in terms of τ . We relate our geometric construction to arithmetic constructions and discuss integrality properties of τ .

Computable obstructions to wait-free computability

Effectively computable obstructions are associated to a distributed decision task (/spl Iscr/,/spl Oscr/,/spl Delta/) in the asynchronous, wait-free, read-write shared-memory model. The key new ingredient of this work is the association of a simplicial complex /spl Tscr/, the task complex, to the input-output relation d. The task determines a simplicial map /spl alpha/ from /spl Tscr/ to the input complex /spl Iscr/. The existence of a wait-free protocol solving the task implies that the map /spl alpha//sub */ induced in homology must surject, and thus elements of H/sub */(/spl Iscr/) that are not in the image of /spl alpha//sub */, are obstructions to solvability of the task. These obstructions are effectively computable when using suitable homology theories, such as mod-2 simplicial homology. We also extend Herlihy and Shavit's Theorem on Spans to the case of protocols that are anonymous relative to the action of a group, provided the action is suitably rigid. For such rigid actions, the quotients of the input complex and the task complex by the group are well-behaved, and obstructions to anonymous solvability of the task are obtained analogously, using the homology of the quotient complexes.

Comparison of graded and ungraded Cousin complexes

Let R = ⊕n∈zRn be a ℤ-graded commutative Noetherian ring and let M be a ℤ-graded R-module. S. Goto and K. Watanabe introduced the graded Cousin complex *C(M)* for M, a complex of graded R-modules. Also one can ignore the grading on M and construct the Cousin complex C(M)* for M, discussed in earlier papers by the second author. The main results in this paper are that *C(M)* can be considered as a subcomplex of C(M)* and that the resulting quotient complex is always exact. This sheds new light on the known facts that, when M is non-zero and finitely generated, C(M)* is exact if and only if *C(M)* is (and this is the case precisely when M is Cohen-Macaulay).

On coalgebras whose complex is a quadratic algebra

We present an approach to the cohomology theory of coalgebras via quotient complexes with quadratic relations. The two types of coalgebras: coassociative and Lie coalgebras are special cases of this theory. This approach could have a possible application to the theory of deformations of algebras.

Triangulation and general position of PL diagrams

We give conditions under which a diagram R ← g P → g Q of polyhedra and PL maps is triangulable. Suppose there is an integer m such that if q 1,…, q r are distinct points of Q with gf -1( q i )∩ gf -1( q i+1 )≠ ∅, i = 1,…, r - 1, then r ⩽ m. Then there are triangulations K, H, and J of P, Q, and R, respectively, with respect to which f and g are simplicial. Moreover, if f and g are surjections, then the relation q ∼ q' if gf -1( q) ∩ gf -1( q') ≠ ∅ defines a quotient complex L of H and simplicial maps ψ : H → L and φ : J → L such that the diagram ▪ is commutative. Suppose R is a PL n-manifold, and for some triangulation T of P, dim σ + max{dim f -1( q) ∩ σ | q ∈ Q}< n for each σ ∈ T. Then g may be approximated by a PL map h : P → R such that the diagram R ← g P → g Q satisfies the hypothesis above with m = 4 n 2, and, hence, is triangulable. These are adaptations and generalizations of results of T. Homma, which we use to give a proof of R. Miller's codimension-three PL approximation theorem: If f : M m → N n , n − m ⩾ 3, is a topological embedding of a PL m-manifold M into a PL n-manifold N, then f can be approximated by PL embeddings.

Cohomological Dimension of Homogeneous Spaces of Complex Lie Groups

In [4] we obtained a formula which represents the completeness of complex Lie groups, and in [3] we generalized it to some types of homogeneous spaces of complex Lie groups. The purpose of the present short note is to give a proof of the generalized formula without the assumption in [3]. Throughout we denote (complex) Lie groups by Roman capital letters and their Lie algebras by the corresponding German small letters respectively. Let (G, H) be a pair of a connected complex Lie group G and a connected closed complex subgroup H of G. Let (K, L) be a pair of maximal compact subgroups K and L of G and H respectively. We consider only such K that contains L. We denote the canonical surjection from g onto the quotient complex vector space g/Ij by 71. Denoting the complex dimension of any complex object X by d(X), we can give the following indices of a G-homogeneous complex manifold G/H:

Dihomology: I. Relations Between Homology Theories

DIHOMOLOGY is a homology theory based on the use of pairs of cells instead of single cells as in classical homology. The pairs considered have some specified geometrical or topological relationship, which is called a facing relation. The resulting double complex is handled by means of spectral sequences. The technique is dual in a certain sense to the taking of tensor products over a ring. For if K, L are A-complexes we feed the A-structure into K®L by forming the quotient complex K® AL. So in dihomology we feed the facing relation into K®L by forming a sub- complex. There is no reason for restricting ourselves to pairs of cells, and the theory extends quite happily to three or more. In fact in the proof of the last theorem in the second paper (12) we have to use a quintuple facing relation.