Let $SL_2$ be the rank one simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. The paper presents a new method for computing the dimension of …
Let $SL_2$ be the rank one simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. The paper presents a new method for computing the dimension of the cohomology spaces $\text{H}^n(SL_2,V(m))$ for Weyl $SL_2$-modules $V(m)$. We provide a closed formula for $\text{dim}\text{H}^n(SL_2,V(m))$ when $n\le 2p-3$ and show that this dimension is bounded by the $(n+1)$-th Fibonacci number. This formula is then used to compute $\text{dim}\text{H}^n(SL_2, V(m))$ for $n=1, 2,$ or $3$. For $n>2p-3$, an exponential bound, only depending on $n$, is obtained for $\text{dim}\text{H}^n(SL_2,V(m))$. Analogous results are also established for the extension spaces $\text{Ext}^n_{SL_2}(V(m_2),V(m_1))$ between Weyl modules $V(m_1)$ and $V(m_2)$. In particular, we determine the degree three extensions for all Weyl modules of $SL_2$. As a byproduct, our results and techniques give explicit upper bounds for the dimensions of the cohomology of the Specht modules of symmetric groups, the cohomology of simple modules of $SL_2$, and the finite group of Lie type $SL_2(p^s)$.
Let $SL_2$ be the rank one simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. The paper presents a new method for computing the dimension of …
Let $SL_2$ be the rank one simple algebraic group defined over an algebraically closed field $k$ of characteristic $p>0$. The paper presents a new method for computing the dimension of the cohomology spaces $\text{H}^n(SL_2,V(m))$ for Weyl $SL_2$-modules $V(m)$. We provide a closed formula for $\text{dim}\text{H}^n(SL_2,V(m))$ when $n\le 2p-3$ and show that this dimension is bounded by the $(n+1)$-th Fibonacci number. This formula is then used to compute $\text{dim}\text{H}^n(SL_2, V(m))$ for $n=1, 2,$ or $3$. For $n>2p-3$, an exponential bound, only depending on $n$, is obtained for $\text{dim}\text{H}^n(SL_2,V(m))$. Analogous results are also established for the extension spaces $\text{Ext}^n_{SL_2}(V(m_2),V(m_1))$ between Weyl modules $V(m_1)$ and $V(m_2)$. In particular, we determine the degree three extensions for all Weyl modules of $SL_2$. As a byproduct, our results and techniques give explicit upper bounds for the dimensions of the cohomology of the Specht modules of symmetric groups, the cohomology of simple modules of $SL_2$, and the finite group of Lie type $SL_2(p^s)$.
In this paper we compute the integral cohomology of the discrete groups SL(2,Z[1/p]), where p is any prime.
In this paper we compute the integral cohomology of the discrete groups SL(2,Z[1/p]), where p is any prime.
In this paper we compute the integral cohomology of the discrete groups SL(2,Z[1/p]), where p is any prime.
In this paper we compute the integral cohomology of the discrete groups SL(2,Z[1/p]), where p is any prime.
We define weak 2-categories of finite dimensional algebras with bimodules, along with collections of operators $\mathbb{O}_{(c,x)}$ on these 2-categories. We prove that special examples $\mathbb{O}_p$ of these operators control all …
We define weak 2-categories of finite dimensional algebras with bimodules, along with collections of operators $\mathbb{O}_{(c,x)}$ on these 2-categories. We prove that special examples $\mathbb{O}_p$ of these operators control all homological aspects of the rational representation theory of the algebraic group $GL_2$, over a field of positive characteristic. We prove that when $x$ is a Rickard tilting complex, the operators $\mathbb{O}_{(c,x)}$ honour derived equivalences, in a differential graded setting. We give a number of representation theoretic corollaries, such as the existence of tight $\mathbb{Z}_+$-gradings on Schur algebras $S(2,r)$, and the existence of braid group actions on the derived categories of blocks of these Schur algebras.
Let k, n be positive integers. Denote by J+k (n) the Lie group of k-jets at 0 of orientation preserving local diffeomorphisms of Rn fixing the origin, with multiplication induced …
Let k, n be positive integers. Denote by J+k (n) the Lie group of k-jets at 0 of orientation preserving local diffeomorphisms of Rn fixing the origin, with multiplication induced by composition of functions. It is a well-known fact that the singular homology of J+k (n) is isomorphic to the singular homology of GL+ (n), the group of n × n matrices with real entries and positive determinant. In this paper we show that the corresponding result is true for the group homotopy of J+k (n) and GL+ (n) as abstract (discrete) groups. In addition, we show that the continuous (and smooth) cohomology groups of J+k (n) and GL+ (n) are also isomorphic as are the Lie algebra cohomology groups of their Lie algebras.
Cette these porte sur la topologie en basse dimension, plus precisement sur la theorie des noeuds. Elle se decompose en deux parties. La premiere consiste en l'etude de l'homologie reduite …
Cette these porte sur la topologie en basse dimension, plus precisement sur la theorie des noeuds. Elle se decompose en deux parties. La premiere consiste en l'etude de l'homologie reduite de Khovanov et ses proprietes combinatoires determinantes pour la calculer. La seconde applique ces outils a la familles des entrelacs toriques. Ces entrelacs dependent uniquement de deux parametres, le nombre de brins et le nombre de tours. Nous calculons l'homologie de tous entrelacs toriques a 3 brins. Pour un nombre de brins fixe et un nombre de tour infini nous montrons que l'homologie correspondante admet la structure supplementaire d'une algebre, que nous calculons precisement pour la famille a 2 brins.
We define a bigraded homology theory whose Euler characteristic is the quantum sl(3) link invariant.
We define a bigraded homology theory whose Euler characteristic is the quantum sl(3) link invariant.
As a generalization of pretzel surfaces, L. Rudolph has introduced a notion of braidzel surfaces in his study of the quasipositivity for pretzel surfaces. In this paper, we show that …
As a generalization of pretzel surfaces, L. Rudolph has introduced a notion of braidzel surfaces in his study of the quasipositivity for pretzel surfaces. In this paper, we show that any oriented link has a braidzel surface. We also introduce a new geometric numerical invariant of links with respect to their braidzel surface and study relationships among them and other “genus” for links.
We determine an $\mathfrak{sl}_2$ module structure on the equivariant Khovanov-Rozanksy homology of T(2,k)-torus links following the framework defined in arXiv:2306.10729.
We determine an $\mathfrak{sl}_2$ module structure on the equivariant Khovanov-Rozanksy homology of T(2,k)-torus links following the framework defined in arXiv:2306.10729.
The aim of this paper is to determine the topology of the variety of representations of the fundamental group of a punctured surface in SL(2,R) with prescribed behavior at the …
The aim of this paper is to determine the topology of the variety of representations of the fundamental group of a punctured surface in SL(2,R) with prescribed behavior at the punctures. In order to do that, we follow the strategy employed by Hitchin in the unpunctured case and we exploit the correspondence between representations of the fundamental group and Higgs bundles in the parabolic case.
We classify all links whose Khovanov homology have ranks no greater than 8, and all three-component links whose Khovanov homology have ranks no greater than 12, where the coefficient ring …
We classify all links whose Khovanov homology have ranks no greater than 8, and all three-component links whose Khovanov homology have ranks no greater than 12, where the coefficient ring is Z/2. The classification is based on the previous results of Kronheimer-Mrowka, Batson-Seed, Baldwin-Sivek, and the authors.
We introduce the adjoint homological Selmer module for an SL$_2$-representation of a knot group, which may be seen as a knot theoretic analogue of the dual adjoint Selmer module for …
We introduce the adjoint homological Selmer module for an SL$_2$-representation of a knot group, which may be seen as a knot theoretic analogue of the dual adjoint Selmer module for a Galois representation. We then show finitely generated torsion-ness of our adjoint Selmer module, which are widely known as conjectures in number theory, and give some concrete examples.
If $\pi :\tilde X \to X$ is a double branched cover, with branching set F, we relate ${H_ \ast }(\tilde X:{\mathbb {Z}_2}),{H_ \ast }(X:{\mathbb {Z}_2}),{H_ \ast }(X,F:{\mathbb {Z}_2})$, and ${H_\ast …
If $\pi :\tilde X \to X$ is a double branched cover, with branching set F, we relate ${H_ \ast }(\tilde X:{\mathbb {Z}_2}),{H_ \ast }(X:{\mathbb {Z}_2}),{H_ \ast }(X,F:{\mathbb {Z}_2})$, and ${H_\ast }(F:{\mathbb {Z}_2})$.
If $\pi :\tilde X \to X$ is a double branched cover, with branching set F, we relate ${H_ \ast }(\tilde X:{\mathbb {Z}_2}),{H_ \ast }(X:{\mathbb {Z}_2}),{H_ \ast }(X,F:{\mathbb {Z}_2})$, and ${H_\ast …
If $\pi :\tilde X \to X$ is a double branched cover, with branching set F, we relate ${H_ \ast }(\tilde X:{\mathbb {Z}_2}),{H_ \ast }(X:{\mathbb {Z}_2}),{H_ \ast }(X,F:{\mathbb {Z}_2})$, and ${H_\ast }(F:{\mathbb {Z}_2})$.
Although computing the Khovanov homology of links is common in literature, no general formulae have been given for all of them. We give the graded Euler characteristic and the Khovanov …
Although computing the Khovanov homology of links is common in literature, no general formulae have been given for all of them. We give the graded Euler characteristic and the Khovanov homology of the 2-strand braid link ,, and the 3-strand braid .
In this note we compute the integral cohomology groups of the subgroups $\Gamma _0(n)$ of $SL(2, \mathbf {Z})$ and the corresponding subgroups $P\Gamma _0(n)$ of its quotient, the classical modular …
In this note we compute the integral cohomology groups of the subgroups $\Gamma _0(n)$ of $SL(2, \mathbf {Z})$ and the corresponding subgroups $P\Gamma _0(n)$ of its quotient, the classical modular group, $PSL(2, \mathbf {Z}).$
Abstract Research mathematicians in algebraic topology will be interested in this new attempt to classify homotopy types of simply connected CW-complexes. This book provides a modern treatment of a long …
Abstract Research mathematicians in algebraic topology will be interested in this new attempt to classify homotopy types of simply connected CW-complexes. This book provides a modern treatment of a long established set of questions in algebraic topology. The author is a leading figure in this important research area.
One way to study the cohomology of a group T of finite virtual cohomological dimension is to find a finite dimensional contractible space X on which F acts properly (such …
One way to study the cohomology of a group T of finite virtual cohomological dimension is to find a finite dimensional contractible space X on which F acts properly (such a space X always exists by [18], 1-7), and to then analyze the action, In this paper we want to consider the arithmetic groups SL2(0) and PSL2(0) = SL2(O)/±I where 0 is the ring of integers in an imaginary quadratic number field fc; the classical choice of X in this case is hyperbolic three-space H, i.e., the associated symmetric space SL2(C)/SU(2). As early as 1892 Bianchi [2] exhibited fundamental domains for the action of PSL2(0) on H for some small values of the discriminant. The space H has also turned out to be very useful in studying the relation between automorphic forms associated to SL2(0) and the cohomology of SL2(O) (cf. [12], [10]), and in studying the topology of certain hyperbolic 3-manifolds (cf, [25]). However, this choice of X is inconvenient for actual explicit computations of the cohomology of r = (P)SL2(0) with integral coefficients because the dimension of H is three, whereas the virtual cohomological dimension of F is two, indicating that it may be possible for F to act properly on a contractible space of dimension two; in addition, the quotient F\H is not compact. A more useful space X for our purposes is given by work of Mendoza [14], which we recall in §3; using Minkowski's reduction theory (cf. §2), he constructs a f-invariant 2-dimensional deformation retract I(fe) of H such that the quotient of I(fc) by any subgroup of F of finite index is compact; I(k) is endowed with a natural CW structure such that the action of F is cellular and the quotient F\I(k) is a finite CW-complex. The main object of this paper is to show how this construction can be used to completely determine the integral homology groups of PSL2(0). This is done by analyzing a spectral sequence which relates the homology of PSL2(0) to the homology of the quotient space PSL2(0)\I(k) and the homology of the stabilizers of the cells (cf. [5], VII). We will confine our computations to the cases where 0 is a euclidean ring, i.e., @-@_d is the ring of integers in fc = Q(V-d) for d = 1,2,3,7 and 11. We will write out in detail the case d = 2 (cf. §5), which contains
Abstract We introduce a refinement of the Bloch-Wigner complex of a field F . This refinement is complex of modules over the multiplicative group of the field. Instead of computing …
Abstract We introduce a refinement of the Bloch-Wigner complex of a field F . This refinement is complex of modules over the multiplicative group of the field. Instead of computing K 2 (F) and K ind 3 (F) - as the classical Bloch-Wigner complex does - it calculates the second and third integral homology of SL 2 (F) . On passing to F × -coinvariants we recover the classical Bloch-Wigner complex. We include the case of finite fields throughout the article.