Joshua Roberts
Given a finitely presented group $G$, Hopf's formula expresses the second integral homology of $G$ in terms of generators and relators. We give an algorithm that exploits Hopf's formula to estimate $H_2(G;k)$, with coefficients in a finite field k, and give examples using $G=SL_2$ over specific rings of integers. These examples are related to a conjecture of Quillen.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we compute the second homology groups of the automorphism group of a free group with coefficients in the abelianization of the free group and its dual group except for 2-torsion part, using combinatorial group theory.
There are different notions of homology and cohomology that can be defined for a group with an action of another group by group automorphisms. In this paper we address three natural questions that arise in this context. Namely, the relation of these notions with the usual (co)homology of a semidirect product, the interpretation of the first homology group as some kind of abelianization and the classification of (invariant) group extensions.
We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a gamma-equivariant G-module A, when a separate group "gamma" acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.
We show the topological Hochschild homology spectrum of a twisted group algebra $\THH(A^τ[G])$ is the Thom spectrum associated to a parametrized orthogonal spectrum $E(A,G)$. We then analyze the structure of the parametrized orthogonal spectrum $E(A,G)$ and show that it is locally trivial.
On the Homology stability range for symplectic groups
本文改进了辛群的整数同调稳定性范围，并揭示了进一步稳定性的障碍与Milnor-Witt K理论的关系。
We study the homology of the dual de Rham complex as functors on the category of abelian groups. We give a description of homology of the dual de Rham complex up to degree 7 for free abelian groups and present a corrected version of the proof of Jean's computations of the zeroth homology group.
We show that a set with an action of a locally finite-dimensional free partially commutative monoid and the corresponding semicubical set have isomorpic homology groups. We build a complex of finite length for the computing homology groups of any asynchronous transition system with finite maximal number of mutually independent events. We give examples of computing the homology groups.
We compute the $G$-equivariant homotopy type of the real topological cyclic homology of spherical group rings with anti-involution induced by taking inverses in the group, where $G$ denotes the group $Gal(\mathbb{C}/\mathbb{R})$. The real topological Hochschild homology of a spherical group ring $\mathbb{S}[Γ]$, with anti-involution as described, is an $O(2)$-cyclotomic spectrum and we construct a map commuting with the cyclotomic structures from the $O(2)$-equivariant suspension spectrum of the dihedral bar construction on $Γ$ to the real topological Hochschild homology of $\mathbb{S}[Γ]$, wh…
We obtain the equivariant K-homology of the classifying space \underline{E}W for W a right-angled or, more generally, an even Coxeter group. The key result is a formula for the relative Bredon homology of \underline{E}W in terms of Coxeter cells. Our calculations amount to the K-theory of the reduced C^*-algebra of W, via the Baum-Connes assembly map.
We define for every dendroidal set X a chain complex and show that this assignment determines a left Quillen functor. Then we define the homology groups $H_n(X)$ as the homology groups of this chain complex. This generalizes the homology of simplicial sets. Our main result is that the homology of X is isomorphic to the homology of the associated spectrum K(X) as discussed in earlier work of the authors. Since these homology groups are sometimes computable we can identify some spectra K(X) which we could not identify before.
In this paper we compute the Bredon homology of wallpaper groups with respect to the family of finite groups and with coefficients in the complex representation ring. We provide explicit bases of the homology groups in terms of irreducible characters of the representation rings of the stabilizers.
For a non-cyclic free group $F$, the second homology of its pronilpotent completion $H_2(\widehat F)$ is not a cotorsion group.
We establish a decomposition of stable homology of automorphism groups of free groups with polynomial contravariant coefficients in term of functor homology. This allows several explicit computations, intersecting results obtained by independent methods by O. Randal-Williams and extending some of them.Our methods rely on the investigation of Kan extensions associated to several categories of free groups, the extension of a cancellation criterium for homology with polynomial coefficients due to Scorichenko, Galatius Theorem identifying the stable homology of automorphism groups of free groups t…
In this paper, we introduce a kind of homology which we call Hawaiian homology to study and classify pointed topological spaces. The Hawaiian homology group has advantages of Hawaiian groups. Moreover, the first Hawaiian homology group is isomorphic to the abelianization of the first Hawaiian group for path-connected and locally path-connected topological spaces. Since Hawaiian homology has concrete elements and abelian structure, its calculations are more routine. Thus we use Hawaiian homology groups to compare Hawaiian groups, and then we obtain some information about Hawaiian groups of some…
A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$, $m<M$, or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps ${\mathbb R}^{m+1} \to {\mathbb R}^{M+1}$ of growing degrees. Also, we find the homology groups of spaces of ${\mathbb Z}_r$-equivariant maps of odd-dimensional spheres for any $r$. As a technical tool, we calculate the homology groups of configurat…
We give a sufficient and necessary condition of the fundamental group homomorphism of a map between manifolds to induce homology equivalences. Moreover, a classification of one-sided h-cobordism of manifolds up to diffeomorphisms is obtained, based on Quillen's plus construction with Whitehead torsions.
For Artin groups of dihedral type, we compute the Bredon homology groups of the classifying space for the family of virtually cyclic subgroups with coefficients in the K-theory of a group ring.
In the paper the Čech border homology and cohomology groups of closed pairs of normal spaces are constructed and investigated. These groups give intrinsic characterizations of Čech homology and cohomology groups based on finite open coverings, homological and cohomological coefficients of cyclicity, small and large cohomological dimensions of remainders of Stone-Čech compactifications of metrizable spaces.
We calculate the homology of the mapping class group in the stable range. The calculation is based on Madsen and Weiss' proof of the "Generalised Mumford Conjecture".
We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute the rational stable homology of the outer automorphism groups and of the mapping class groups of the associated aspherical nil-manifolds in the TOP, PL, and DIFF categories.
We show in this work that homology in degree d of a congruence group, in a very general framework, defines a weakly polynomial functor of degree at most 2d and we describe this functor modulo polynomial functors of smaller degree. Our main tool is a spectral sequence connecting homology of congruence-like groups (in a formal setting close to the one introduced with Vespa in 2010 for orthogonal groups) and functor homology. We prove and use in a crucial way properties of some tensor structures and derived Kan extensions on polynomial functors.Our results extend especially, with different method…
We compute the mod 2 homology of spin mapping class groups in the stable range. In earlier work we computed the stable mod p homology of the oriented mapping class group, and the methods and results here are very similar. The forgetful map from the spin mapping class group to the oriented mapping class groups induces a homology isomorphism for odd p but for p=2 it is far from being an isomorphism. We include a general discussion of tangential structures on 2-manifolds and their mapping class groups and then specialise to spin structures. As in Madsen and Weiss' solution to Mumford's conjecture…
In this paper, we consider homology groups induced by the exterior algebra generated by a simplicial compliment of a simplicial complex $K$. These homology groups are isomorphic to the Tor-groups $\mathrm{Tor}_{i, J}^{\mathbf{k}[m]}(\mathbf{k}(K),\mathbf{k})$ of the face ring $\mathbf{k}(K)$, which is very useful and much studied in toric topology. By using $\check{C}ech$ homology theory and Alexander duality theorem, we prove that these homology groups have dualities with the simplicial cohomology groups of the full subcomplexes of $K$. Then we give a new proof of Hochster's theorem.
We consider the Birman-Hilden inclusion $φ\colon\mathfrak{Br}_{2g+1}\toΓ_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $φ$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(Σ_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in $φ^*(H_1(Σ_{g,1}))$ has only $4$-torsion.
In this paper, we deal with stable homology computations with twisted coefficients for mapping class groups of surfaces and of 3-manifolds, automorphism groups of free groups with boundaries and automorphism groups of certain right-angled Artin groups. On the one hand, the computations are led using semidirect product structures arising naturally from these groups. On the other hand, we compute the stable homology with twisted coefficients by FI-modules. This notably uses a decomposition result of the stable homology with twisted coefficients for pre-braided monoidal categories proved in this …
We give a proof of the Jardine-Tillmann generalized group completion theorem. It is much in the spirit of the original homology fibration approach by McDuff and Segal, but follows a modern treatment of homotopy colimits, using as little simplicial technology as possible. We compare simplicial and topological definitions of homology fibrations.
The aim of this paper is to investigate the homology groups of mathematical models of concurrency. We study the Baues-Wirsching homology groups of a small category associated with a partial monoid action on a set. We prove that these groups can be reduced to the Leech homology groups of the monoid. For a trace monoid with an action on a set, we will build a cubical complex of free Abelian groups with homology groups isomorphic to the integral homology groups of the action category. It allows us to solve the problem posed by the author in 2004 of the constructing an algorithm for computing homo…
We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of p-local finite groups. We prove that the cohomology ring of Robinson's construction is in some precise sense very close to the cohomology ring of the fusion system it realizes.
Homology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus $g$ with one boundary component, that are double coverings of the disk ramified over $n = 2g + 1$ points. The main part of such homology is described by the homology of the braid group with coefficients in a symplectic representation, namely the braid group $\mathrm{Br}_n$ acts on the first homology group of a genus $g$ surface via Dehn twists. Our computations shows that such groups have only $2$-torsion. We also inves…
Stable rational homology of the IA-automorphism groups of free groups
研究自由群IA自同构群的有理同调的稳定范围，确定其Albanese同调的下界，并计算n≥9时的第三Albanese同调。
We study homology groups of elementary Petri nets for the pipeline systems. We show that the integral homology groups of these nets in dimensions 0 and 1 equal the group of integers, and they are zero in other dimensions. We prove that directed homology groups of elementary Petri nets are zero in all dimensions.
Homotopy type through homology groups
本文证明了具有两个连续维数自由有限生成约化同调且其余维数同调平凡的复形必同伦等价于相应维数球面的楔和，并推广到其他维数对。
Hyperoctahedral homology is the homology theory associated to the hyperoctahedral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyperoctahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the …
We give a Dehn-Nielsen type theorem for the homology cobordism group of homology cylinders by considering its action on the acyclic closure, which was defined by Levine, of a free group. Then we construct an additive invariant of those homology cylinders which act on the acyclic closure trivially. We also describe some tools to study the automorphism group of the acyclic closure of a free group generalizing those for the automorphism group of a free group or the homology cobordism group of homology cylinders.
Such modern applications of topology as data analysis and digital image analysis have to deal with noise and other uncertainty. In this environment, topological spaces often appear equipped with a real valued function. Persistence is a measure of robustness of the homology classes of the filtration of the lower level sets of this function. In this paper we introduce the homology group of filtration as the product of the kernels of the homology maps of the inclusions. This group contains all possible homology classes in all elements of the filtration so that we can later pick the features that …
In this paper we compute the homology of the braid groups, with coefficients in the module Z[q^+-1] given by the ring of Laurent polynomials with integer coefficients and where the action of the braid group is defined by mapping each generator of the standard presentation to multiplication by -q. The homology thus computed is isomorphic to the homology with constant coefficients of the Milnor fiber of the discriminantal singularity.
In this paper, we compute the second mod $2$ homology of an arbitrary Artin group, without assuming the $K(π,1)$ conjecture. The key ingredients are (A) Hopf's formula for the second integral homology of a group and (B) Howlett's result on the second integral homology of Coxeter groups.
Homology of the group Aut(F_n) of automorphisms of a free group on n generators is known to be independent of n in a certain stable range. Using tools from homotopy theory, we prove that in this range it agrees with homology of symmetric groups. In particular, Aut(F_n) has no stable rational homology, settling a conjecture of Hatcher and Vogtmann.
Symmetric homology is an analog of cyclic homology in which the cyclic groups are replaced by symmetric groups. The foundations for the theory of symmetric homology of algebras are developed in the context of crossed simplicial groups using derived functors and the symmetric bar construction of Fiedorowicz. The symmetric homology of group rings is related to stable homotopy theory. Two chain complexes are constructed that compute symmetric homology, as well as two spectral sequences. In the setup of the second spectral sequence, a complex isomorphic to the suspension of the cycle-free chessboa…
Let $Σ_g$ denote the closed orientable surface of genus $g$ and fix an arbitrary simplicial triangulation of $Σ_g$. We construct and study a natural surjective group homomorphism from the surface braid group on $n$ strands on $Σ_g$ to the first singular homology group of $Σ_g$ with integral coefficients. In particular, we show that the kernel of this homomorphism is generated by canonical braids which arise from the triangulation of $Σ_g$. This provides a simple description of natural subgroups of surface braid groups which are closely tied to the homology groups of the surfaces $Σ_g$.
We consider the torsion of homology groups of right pointed sets over a partially commutative monoid M(E,I)
We study the homology of pointed sets over a partially commutative monoid.
In this note, we outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed simplicial groups and the homological algebra of module-valued functors. The symmetric homology of group algebras is related to stable homotopy theory. Two spectral sequences for computing symmetric homology are constructed. The relation to cyclic homology is discussed and some conjectures and questions towards further work are discussed.
Abelian covers of surfaces and the homology of the Torelli group
本文研究了映射类群和 Torelli 群在曲面泛阿贝尔覆盖有理同调系数下的第一同调群，证明了单边界与单穿孔情形下维度的有限与无限对比性，为计算 Torelli 群第二有理同调群提供关键支撑。
Let $H_{g}$ be a 3-dimensional handlebody of genus $g$. We determine the twisted first homology group of the mapping class group of $H_{g}$ with coefficients in the first integral homology group of the boundary surface $\partial H_{g}$ for $g\geq 2$.
The second rational homology of the Torelli group
本文计算了亏格 $g \geq 6$ 时 Torelli 群的二阶有理同调群。
We define a homology for ternary groups using both associativity and skew elements. We describe the odd-even construction which yields many examples of ternary groups. We define the ternary knot group, consider its homomorphisms into ternary groups, and discuss the applications.
We calculate rational homology groups of spaces of non-resultant (i.e. having no non-trivial common zeros) systems of homogemeous quadratic polynomials in R^3
In the early 1980s Ross Geoghegan and I calculated the homology of Richard Thompson's group F as a graded abelian group. It turns out that the homology admits a natural ring structure, which I calculate in this paper. As a byproduct, I obtain the ring structure on the cohomology.
We use covering space theory and the fundamental group of complements of complexified-real line arrangements to explore the associated Milnor fiber. This work yields a combinatorially determined upper bound on the rank of the first homology group of the Milnor fiber. Under certain combinatorial conditions, we then show that one may determine the exact rank of the group and show that it is torsion free.
An integral homology theory on the category of undirected reflexive graphs was constructed in [2]. A geometrical method to understand behaviors of $1$- and $2$-simplices under differential maps of the theory was developed in [3] and led us to an independent proof that the first homology group of any cycle graphs is $\mathbb{Z}$, as it was proved before by a version of Hurewicz theorem harshly defined and shown in [1] and [2]. In this work, we use the old method in [3] to study behaviors of the first homology group of Hamiltonian graphs. We discovered that $H_1(G)$ is torsion-free, for any Hami…
A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their homology cobordism groups have infinitely generated abelian quotient groups by using Reidemeister torsion invariants. In this paper, we first investigate their abelian quotients again by using another invariant called the Magnus representation. After that, we apply the machinery …
We use methods from the cohomology of groups to describe the finite groups which can act freely and homologically trivially on closed 3-manifolds which are rational homology spheres.
We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,\mathbb{Z})$-module.
Let G be a finite group acting freely on a compact oriented surface S by homeomorphisms preserving the orientation. Then, there exists a G-invariant Lagrangian subspace in the first homology group of S.
Given a genus-g Heegaard splitting of a 3-sphere, the genus-g Goeritz group is defined to be the group of the isotopy classes of orientation preserving homeomorphism of the 3-sphere that preserve the splitting. In this paper, we determine the twisted first (co)homology group of the genus-2 Goeritz group of 3-sphere.
In this article, we computed the homology groups of real Grassmann manifold $G_{n,m}(\mathbb{R})$ by Witten complex.
Using fixed-point-free group actions, we set up a scheme to define nested classes of groups indexed over ordinals. Restricting to cellular actions on CW-complexes, we find new classes as well as new characterizations for some well-known classes, such as virtually polycyclic groups. We generalize properties of the virtual cohomological dimension of a group to groups with jump (co)homology and prove that a core subclass of a new class of groups has jump (co)homology.
If $Q$ is a group acting as a group of automorphisms of another group $G$ (with finite orbits), denote by $C_*(G)^Q$ the subcomplex of $Q$-invariant chains in the bar complex $C_*(G)$. In this paper, we study the homology of the complex $C_*(G)^Q$ and compute several examples.
The second integral homology of even Artin groups
本文用Hopf公式计算偶数Artin群的第二整同调，并应用于杯积、Pontryagin积及偶数Coxeter群的同调计算。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。