Takao Satoh
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.
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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.
Topological $ΔG$ homology of rings with twisted $G$-action
本文提出了一种新的拓扑同调理论，用于研究具有扭G作用的环结构，并证明了其在循环空间上的等价性。
We study twisted $Spin^c$-manifolds over a paracompact Hausdorff space $X$ with a twisting $α: X \to K(\ZZ, 3)$. We introduce the topological index and the analytical index on the bordism group of $α$-twisted $Spin^c$-manifolds over $(X, α)$, taking values in topological twisted K-homology and analytical twisted K-homology respectively. The main result of this paper is to establish the equality between the topological index and the analytical index. We also define a notion of geometric twisted K-homology, whose cycles are geometric cycles of $(X, \a)$ analogous to Baum-Douglas's geometric cycl…
We prove that the twisted K-homology of a simply connected simple Lie group G of rank n is an exterior algebra on n-1 generators tensor a cyclic group. We give a detailed description of the order of this cyclic group in terms of the dimensions of irreducible representations of G and show that the congruences determining this cyclic order lift along the twisted index map to relations in the twisted Spin-c bordism group of G.
Twisted topological Hochschild homology of $C_n$-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work of Hill, Hopkins, and Ravenel on norms in equivariant homotopy theory. In this paper we introduce tools for computing twisted THH, which we apply to computations for Thom spectra, Eilenberg-MacLane spectra, and the real bordism spectrum $MU_{\mathbb{R}}$. In particular, we construct an equivariant version of the Bökstedt spectral sequence, the formulation of which requires further development of the Hochschild homology of Green f…
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…
We study the twisted Novikov homology of the complement of a complex hypersurface in general position at infinity. We give a self-contained topological proof of the vanishing (except possibly in the middle degree) of the twisted Novikov homology groups associated to positive cohomology classes of degree one defined on the complement.
The purpose of this note is to clarify some details in McDuff and Segal's proof of the group-completion theorem and to generalize both this and the homology fibration criterion of McDuff to homology with twisted coefficients. This will be used in forthcoming work to identify the limiting homology of "oriented" configuration spaces, which doubly cover the classical configuration spaces of distinct unordered points in a manifold.
The algebraic structure of twisted topological Hochschild homology
本文研究扭曲THH的代数结构，计算了实bordism谱的$C_2$-扭曲THH，并扩展了其到扭曲Bökstedt谱序列，证明其为交换$\underline{E}_\star(R)$-代数。
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 …
Equivariant Witt Complexes and Twisted Topological Hochschild Homology
本文研究扭曲拓扑霍奇同调（THH）的等变同伦群代数结构，定义等变Witt复形并证明其捕捉扭曲THH等变同伦的结构，为等变代数研究提供新贡献。
We study in this article stable homology of automorphism groups of free groups with coefficients twisted by a poynomial functor. We show that this homology is zero for a reduced covariant polynomial functor. For a reduced contravariant functor, we compute the first homology group, which is in general non zero. Our methods relie on the use of functor categories. ---On étudie dans cet article l'homologie stable des groupes d'automorphismes des groupes libres à coefficients tordus par un foncteur polynomial. On montre que cette homologie est nulle pour un foncteur polynomial covariant réduit. Dan…
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.
Szczarba's twisted shuffle and equivariant path homology of directed graphs
本文研究了标记设定下路径同调的等变性质，通过扭曲洗牌定义了链同构，并应用于有向图的Borel构造。
Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $\mathsf V$, as well as for small $\mathsf V$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homolo…
Let $A$ be either a simplicial complex $K$ or a small category $\mathcal C$ with $V(A)$ as its set of vertices or objects. We define a twisted structure on $A$ with coefficients in a simplicial group $G$ as a function $$ δ\colon V(A)\longrightarrow \operatorname{End}(G), \quad v\mapsto δ_v $$ such that $δ_v\circ δ_w=δ_w\circ δ_v$ if there exists an edge in $A$ joining $v$ with $w$ or an arrow either from $v$ to $w$ or from $w$ to $v$. We give a canonical construction of twisted simplicial group as well as twisted homology for $A$ with a given twisted structure. Also we determine the homotopy t…
We prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the general linear groups over the integers and the automorphism groups of free groups. The proof presented here uses a general result that applies to arbitrary extensions of groups, and that has other applications as well.
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 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.
We compute the homology groups $H_*(\operatorname{Out}(F_7);\mathbb Q)$ of the outer automorphism group of the free group of rank $7$. We produce in this manner the first rational homology classes of $\operatorname{Out}(F_n)$ that are neither constant ($*=0$) nor Morita classes ($*=2n-4$).
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。
Given a group $G$ of automorphisms of a matroid $M$, we describe the representations of $G$ on the homology of the independence complex of the dual matroid $M^*$. These representations are related with the homology of the lattice of flats of $M$, and (when $M$ is realizable) with the top cohomology of a hyperplane arrangement. Finally we analyze in detail the case of the complete graph, which has applications to algebraic geometry.
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 $g\geq 2$.
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.
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.
Twisted homology operations for $E_{\infty}$-algebras
本文发展了$E_{\infty}$-代数扭曲同调上的运算理论，通过在函子范畴中定义扭曲版Dyer--Lashof运算与乘积，并给出自由$E_{\infty}$-代数同调的显式基，证明了这些运算生成所有扭曲同调运算。
The stable Albanese homology of the IA-automorphism groups of free groups
本文研究了自由群IA-自同构群的Albanese同调，确定了其在n≥3时的表示结构，并揭示了其与外自同构群及Aut(Fn)稳定上同调的关系。
This work concerns the Koszul complex $K$ of a commutative noetherian local ring $R$, with its natural structure as differential graded $R$-algebra. It is proved that under diverse conditions, involving the multiplicative structure of $H(K)$, any dg $R$-algebra automorphism of $K$ induces the identity map on $H(K)$. In such cases, it is possible to define an action of the automorphism group of $R$ on $H(K)$. On the other hand, numerous rings are described for which $K$ has automorphisms that do not induce the identity on $H(K)$. For any $R$, it is shown that the group of automorphisms of $H(K)…
We show that the equivariant chain complex associated to a minimal CW-structure X on the complement M(A) of a hyperplane arrangement A, is independent of X. When A is a sufficiently general linear section of an aspheric arrangement, we explain a new way for computing the twisted homology of M(A).
Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology
本研究计算了带边紧致可定向曲面映射类群以单位切丛同调为系数的稳定扭曲上同调，发现其为协变函子超出传统框架，还计算了符合传统框架的一阶上同调系数的稳定上同调群。
We study the classifying space of a twisted loop group $L_σG$ where $G$ is a compact Lie group and $σ$ is an automorphism of $G$ of finite order modulo inner automorphisms. Equivalently, we study the $σ$-twisted adjoint action of $G$ on itself. We derive a formula for the cohomology ring $H^*(BL_σG)$ and explicitly carry out the calculation for all automorphisms of simple Lie groups. More generally, we derive a formula for the equivariant cohomology of compact Lie group actions with constant rank stabilizers.
We study the cohomology of Aut(F_n) and Out(F_n) with coefficients in the modules \wedge^q H, \wedge H^*, Sym^q H or Sym^q H^*, where H is the Out(F_n)-module obtained by abelianising the free group F_n. For reasons which are not conceptually clear, taking coefficients in H and its related modules behaves in a far less trivial way than taking coefficients in H^* and its related modules. Based on a conjectural homology stability theorem for spaces of graphs labeled by a simply connected background space, we give a stable integral calculation of these groups in low degrees, and modulo a further …
We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra of a k-graph by a T-valued 2-cocycle and demonstrate that examples include all noncommutative tori. In t…
On Realisability of Twisted Homology
探讨流形\(X\)扭曲整数系数同调或上同调类由子流形实现的问题，引入扭曲配边类和空间，通过构造参数化波斯特尼科夫塔推导障碍，给出非可定向流形中不可实现整数同调类的例子。
Automorphisms of Frobenius twisted de Rham cohomology
本文证明Frobenius扭转的de Rham上同调函子的自同构模空间为$\mathbb{G}_m$，并利用该作用重新证明了Bhatt-Petrov-Vologodsky的分解定理。
Twisted homology stability of O_n for valuation rings
本文扩展Vogtmann的论证，展示当A为满足算术条件的赋值环时，欧几里得正交群O_n(A)的同调稳定性。特别地，若A为Henselian赋值环且其余数域具有有限的Pythagoras数，则O_n(A)表现出同调稳定性。结果包括Vogtmann的成果，并适用于各种扭曲系数。
We show that the number of twisted conjugacy classes is infinite for any automorphism of non-elementary, Gromov hyperbolic group . An analog of Selberg theory for twisted conjugacy classes is proposed.
Scissors automorphism groups I: Homological stability and K-theory
本文证明在温和条件下剪刀自同构群的同调独立于对象，可由剪刀同余 K-理论谱表达，提供群论解释与计算方法，并在经典情形及后续中得出同调计算与结构结果。
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.
Twisted Morse complexes
本文研究闭有限维光滑流形上的扭曲Morse同调与上同调，证明相关定理与多类同调/上同调的同构关系，并通过多个计算实例验证了扭曲Morse复形的有效性。
We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected sum of l closed almost complex manifolds necessarily an almost complex manifold?". We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on (n-1)-connected 2n-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexisten…
We show that the homology of the automorphism group of a right-angled Artin group stabilizes under taking products with any right-angled Artin group.
Let $Σ$ be a compact oriented surface. The Dehn twist along every simple closed curve $γ\subset Σ$ induces an automorphism of the fundamental group $π$ of $Σ$. There are two possible ways to generalize such automorphisms if the curve $γ$ is allowed to have self-intersections. One way is to consider the `generalized Dehn twist' along $γ$: an automorphism of the Malcev completion of $π$ whose definition involves intersection operations and only depends on the homotopy class $[γ]\in π$ of $γ$. Another way is to choose in the usual cylinder $U:=Σ\times [-1,+1]$ a knot $L$ projecting onto $γ$, to p…
We show in this article that, for any group $G$ indecomposable for the free product * and non-isomorphic to $\mathbf{Z}$, the canonical inclusion ${\rm Aut}(G^{*n})\to {\rm Aut}(G^{* n+1})$ induces an isomorphism between the homology groups $H_i$ for $n\geq 2i+2$, as was conjectured by Hatcher and Wahl. In fact we show a little more --- in particular, the result is true for any group $G$ if we replace the automorphism group of the free product by the subgroup of symmetric automorphisms. For this purpose we use constructions and acyclicity results due to McCullough-Miller and Chen-Glover-Jensen…
The observation that a graph of rank $n$ can be assembled from graphs of smaller rank $k$ with $s$ leaves by pairing the leaves together leads to a process for assembling homology classes for $Out(F_n)$ and $Aut(F_n)$ from classes for groups $Γ_{k,s}$, where the $Γ_{k,s}$ generalize $Out(F_k)=Γ_{k,0}$ and $Aut(F_k)=Γ_{k,1}$. The symmetric group $Σ_s$ acts on $H_*(Γ_{k,s})$ by permuting leaves, and for trivial rational coefficients we compute the $Σ_s$-module structure on $H_*(Γ_{k,s})$ completely for $k \leq 2$. Assembling these classes then produces all the known nontrivial rational homology …
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