Petr M. Akhmet'ev
Collection of (equivariant) $\rm{PL}$-mappings admitting a relative abelian, cyclic, quaternionic, bicyclic, and quaternionic-cyclic structures are constructed.
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On the Homotopy Groups of the Suspended Quaternionic Projective Plane and Applications
本文计算了悬浮四元数射影平面同伦群在特定范围内的2、3分量，并应用于局部化悬浮四元数射影空间的CW复形分类及自smash积分解。
The biquotient model of the total space of the projectivization quaternionic bundles over certain manifolds
本文计算四元数流形上四元数切丛诱导的射影丛全空间的有理同伦型，并给出底空间为四元数射影空间时的双商模型表示。
Tautological rings of fake quaternionic spaces
本文结合 Berglund 的有理同伦论方法与族符号差定理，证明几乎所有同伦等价于 ℍP² 的流形的重言环 Krull 维数均为零。
The notion of a cyclic map g: A -> X is a natural generalization of a Gottlieb element in pi_n(X). We investigate cyclic maps from a rational homotopy theory point of view. We show a number of results for rationalized cyclic maps which generalize well-known results on the rationalized Gottlieb groups.
Symmetries of exotic spheres via complex and quaternionic Mahowald invariants
本文利用新的同调工具证明了无限族exotic球面存在光滑U(1)和Sp(1)作用，核心方法是复数和四元数马霍瓦德不变量，主要贡献是证明了该不变量将稳定茎中的元素映射到更高稳定茎中的元素。
A long-standing conjecture of Farrell and Zdravkovska and independently S.~T.~Yau states that every almost flat manifold is the boundary of a compact manifold. This paper gives a simple proof of this conjecture when the holonomy group is cyclic or quaternionic. The proof is based on the interaction between flat bundles and involutions.
We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\widehat{\mathcal O}$, which describes modules over $\mathcal O$ with invariant inner products. We show that $\widehat{\mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\widehat{\mathcal O}$ in terms of derivations and module maps. As an application we construct a homotopy inner product over the commutative operad on the cochains of any Poincaré duali…
We introduce the notion of homotopy inner products for any cyclic quadratic Koszul operad $\mathcal O$, generalizing the construction already known for the associative operad. This is done by defining a colored operad $\hat{\mathcal O}$, which describes modules over $\mathcal O$ with invariant inner products. We show that $\hat{\mathcal O}$ satisfies Koszulness and identify algebras over a resolution of $\hat{\mathcal O}$ in terms of derivations and module maps. An application to Poincaré duality on the chain level of a suitable topological space is given.
We show that the tensor product of two cyclic $A_\infty$-algebras is, in general, not a cyclic $A_\infty$-algebra, but an $A_\infty$-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the pairahedra, which are contractible polytopes controlling the combinatorial structure of an $A_\infty$-algebra with homotopy inner products, and use it to define a categorically closed tensor product. A cyclic $A_\infty$-algebra can be thought of as an $A_\infty$-algebra with homotopy inner products whose higher inner products are trivial. However, the highe…
Equivariant rigidity of complex and quaternionic moment--angle manifolds
本文研究复和四元数时刻-角流形的等价拓扑刚性，通过将其分类转化为其拟 torus 商的等价刚性及关联主丛的分类，证明复时刻-角流形的等价刚性，并在四元数情形中对四维拟 torus 商的流形建立完全等价刚性，基于四次特征类提供更高维的主刚性陈述。
In part I of this work we studied the spaces of real algebraic cycles on a complex projective space P(V), where V carries a real structure, and completely determined their homotopy type. We also extended some functors in K-theory to algebraic cycles, establishing a direct relationship to characteristic classes for the classical groups, specially Stiefel-Whitney classes. In this sequel, we establish corresponding results in the case where V has a quaternionic structure. The determination of the homotopy type of quaternionic algebraic cycles is more involved than in the real case, but has a simi…
In the present paper the cyclic homology functor from the category of $A_\infty$-algebras over any commutative unital ring $K$ to the category of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy equivalences of $A_\infty$-algebras into isomorphisms of graded modules. As a corollary, it is obtained that the cyclic homology of an $A_\infty$-algebra over any field is isomorphic to the cyclic homology of the $A_\infty$-algebra of homologies for the source $A_\infty$-algebra.
On the structure of the $RO(G)$-graded homotopy of $H\underline{M}$ for cyclic $p$-groups
本文研究循环p群上Eilenberg-MacLane谱的RO(G)-分次同伦Mackey函子结构，通过定义定向类并证明归纳定理，计算了若干关键正负锥，为切片谱序列提供基础。
In this paper we apply homotopical localization to the framework of differential graded algebras over an operad. We get plus construction by performing nullification with respect to an universal acyclic algebra. This plus construction for Lie and Leibniz algebras up to homotopy yields two new homology theories for associative algebras. They are closely related to cyclic and Hochschild homologies. We also compute their first homology groups.
A Positive Answer to a Question of K. Borsuk on the Capacity of Polyhedra with Finite by Cyclic Fundamental Group
该研究针对Borsuk关于有限多面体支配形状数量的问题，证明了具有有限被循环扩张基本群（含秩1阿贝尔基本群）的多面体仅支配有限个同伦型，还证明多面体仅支配有限个单连通CW复形同伦型，给出相关问题的部分肯定解答。
We define in the setting of homotopy type theory an H-space structure on $\mathbb S^3$. Hence we obtain a description of the quaternionic Hopf fibration $\mathbb S^3\hookrightarrow\mathbb S^7\twoheadrightarrow\mathbb S^4$, using only homotopy invariant tools.
We construct a discrete model of the homotopy theory of $S^1$-spaces. We define a category $\sP$ with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. $\sP$ inherits a model structure from the model structures on the categories of simplicial sets and cyclic sets. We then show that there is a Quillen equivalence between $\sP$ and the model category of $S^1$-spaces in which weak equivalences and fibrations are maps inducing weak equivalences and fibrations on passage to all fixed point sets.
An action of A on X is a map F: AxX to X such that F|_X = id: X to X. The restriction F|_A: A to X of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which has been studied extensively. We prove some general results about actions and their Eckmann-Hilton duals. For instance, we classify the actions on an H-space that are compatible with the H-structure. As a corollary, we prove that if any two actions F and F' of A on X have cyclic maps f and f' with Omega(f) = Omega(f'), then Omega(F) and Omega(F') give the sam…
A solution to the Kervaire invariant problem is presented. We introduce the concepts of abelian structure on skew-framed immersions, bicyclic structure on $\Z/2^{[3]}$--framed immersions, and quaternionic-cyclic structure on $\Z/2^{[4]}$--framed immersions. Using these concepts, we prove that for sufficiently large $n$, $n=2^{\ell}-2$, an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant. Additionally, for $\ell \ge 12$ (i.e., for $n \ge 4094$) an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant if this skew-frame…
We show that Connes' B-operator on a cyclic differential graded k-module M is a model for the canonical circle action on the geometric realization of M. This implies that the negative cyclic homology and the periodic cyclic homology of a differential graded category can be identified with the homotopy fixed points and the Tate fixed points of the circle action on its Hochschild complex.
In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization which involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Boekst…
We introduce a strong homotopy notion of a cyclic symmetric inner product of an A-infinity algebra and prove a characterization theorem in the formalism of the infinity inner products by Tradler. We also show that it is equivalent to the notion of a non-constant symplectic structure on the corresponding formal non-commutative supermanifold. We show that (open Gromov-Witten type) potential for a cyclic filtered A-infinity algebra is invariant under the cyclic filtered A-infinity homomorphism up to reparametrization, cyclization and a constant addition, generalizing the work of Kajiura.
Using the $E_\infty-$structure on singular cochains, we construct a homotopy coherent map from the cyclic bar construction of the differential graded algebra of cochains on a space to a model for the cochains on its free loop space. This fills a gap in the paper "Cyclic homology and equivariant homology" by John D.S. Jones.
We investigate the exponents of the total Cohen groups $[Ω(\mathbb S^{r+1}), Ω(Y)]$ for any $r\ge 1$. In particular, we show that for $p\ge 3$, the $p$-primary exponents of $[Ω(\mathbb S^{r+1}), Ω(\mathbb S^{2n+1})]$ and $[Ω(\mathbb S^{r+1}), Ω(\mathbb S^{2n})]$ coincide with the $p$-primary homotopy exponents of spheres $\mathbb S^{2n+1}$ and $\mathbb S^{2n}$, respectively. We further study the exponent problem when $Y$ is a space with the homotopy type of $Σ(n)/G$ for a homotopy $n$-sphere $Σ(n)$, the complex projective space $\mathbb{C}P^n$ for $n\ge 1$ or the quaternionic projective space …
Dwyer--Kan homotopy theory for cyclic operads
该研究提出对称幺半基范畴上着色循环 operad 的通用定义，借助其到着色 operad 的遗忘函子的左右伴随显式公式，将单纯集富集着色 operad 范畴的 Cisinski--Moerdijk 模型结构提升到对应循环 operad 范畴。
Fibrewise topological complexity of sphere and projective bundles
本文建立了球丛逐纤维拓扑复杂度的稳定同伦理论版本，推广了Farber和Weinberger的结果，并用平行方法对实、复、四元数射影丛证明了类似结论，同时探讨了Farber和Grant的对称化不变量。
Geometric approach to stable homotopy groups of spheres II; Arf-Kervaire Invariants
本文通过几何方法解决球面稳定同调群中的凯瓦里不变量问题，引入了新的对称结构并证明了相关定理。
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…
The Bredon equivariant cohomology of a point for cyclic groups
本文研究了奇数阶循环群下点的Bredon等变上同调，通过代数方法将其转化为稳定球面同调群问题。
In this paper, we study the equivariant homotopy type of a connected sum of linear actions on complex projective planes defined by Hambleton and Tanase. These actions are constructed for cyclic groups of odd order. We construct cellular filtrations on the connected sum using spheres inside unitary representations. A judicious choice of filtration implies a splitting on equivariant homology for general cyclic groups under a divisibility hypothesis, and in all cases for those of prime power order.
The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (i.e., the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-dimensional handlebodies with marked discs on their boundaries. A modification of the argument provides a new and elementary proof of K. Costello's theorem that the derived modular envelope of the associative operad is homotopy equivalent to the ``open string'' modular operad made …
We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several non-trivial natural transformations from algebraic K-theory to THH(-). In an intermediate step, we prove that the homotopy theory of dg categories is Quillen equivalent, through a four step zig-zag of Quillen equivalences, to the homotopy theory of Eilenberg-Mac Lane spectral categories. Finally, we show that over the rationals two dg categories are topological equivalent if and only if they are quasi-equivalent.
In this paper we compute $RO(G)$-graded homotopy Mackey functors of $H\underline{\mathbb{Z}}$, the Eilenberg-Mac Lane spectrum of the constant Mackey functor of integers for cyclic p-groups and give a complete computation for $G = C_{p^2}$ . We also discuss homological algebra of $\underline{\mathbb{Z}}$-modules for cyclic $p$-groups, and interactions between these two. The goal of this paper is to understand various slice spectral sequences as $RO(G)$-graded spectral sequences of Mackey functors.
Cyclic operads through modules
本文提出通过模计算循环operad映射空间的方法，并应用于计算循环Batalin-Vilkovisky（Hopf余）operad的同伦自同构空间。
First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic $A_\infty$ algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)-operad of CW complexes whose constituent spaces form a homotopy associative version of the Cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyc…
We compute the $\mathbb{A}^1$-localization of several invariants of schemes namely, topological Hochschild homology ($\mathrm{THH}$), topological cyclic homology ($\mathrm{TC}$) and topological periodic cyclic homology ($\mathrm{TP}$). This procedure is quite brutal and kills the completed versions of most of these invariants. The main ingredient for the vanishing statements is the vanishing of $\mathbb{A}^1$-localization of de Rham cohomology (and, eventually, crystalline cohomology) in positive characteristics.
First we argue that many BV and homotopy BV structures, including both familiar and new examples, arise from a common underlying construction. The input of this construction is a cyclic operad along with a cyclically invariant Maurer-Cartan element in an associated Lie algebra. Using this result we introduce and study the operad of cyclically invariant operations, with instances arising in cyclic cohomology and $S^1$ equivariant homology. We compute the homology of the cyclically invariant operations; the result being the homology operad of $\mathcal{M}_{0,n+1}$, the uncompactified moduli spac…
We propose a construction of an analogue of the Hill-Hopkins-Ravenel relative norm $N_{H}^{G}$ in the context of a positive dimensional compact Lie group $G$ and closed subgroup $H$. We explore expected properties of the construction. We show that in the case when $G$ is the circle group (the unit complex numbers), the proposed construction here agrees with the relative norm constructed by Angeltveit, Gerhardt, Lawson, and the authors using the cyclic bar construction. Our construction is based on a new perspective on equivariant factorization homology, using framings to convert from actions o…
Classical work of Lee, Schultz, and Stolz relates the smooth transformation groups of exotic spheres to the stable homotopy groups of spheres. In this note, we apply recent progress on the latter to produce smooth circle and cyclic group actions on certain exotic spheres.
We investigate the combinatorial data arising from the classification of equivariant homotopy commutativity for cyclic groups of order $G=C_{p_1 \cdots p_n}$ for $p_i$ distinct primes. In particular, we will prove a structural result which allows us to enumerate the number of $N_\infty$-operads for $C_{pqr}$, verifying a computational result.
In this paper one considers three homotopy functors on the category of manifolds, $hH^\ast, cH^\ast, sH^\ast,$ and parallel them with other three homotopy functors on the category of connected commutative differential graded algebras, $HH^\ast, CH^\ast, SH^\ast.$ If $P$ is a smooth 1-connected manifold and the algebra is the de-Rham algebra of $P$ the two pairs of functors agree but in general do not. The functors $ HH^\ast $ and $CH^\ast$ can be also derived as Hochschild resp. cyclic homology of commutative differential graded algebra, but this is not the way they are introduced here. The th…
Let $M$ be a closed, oriented manifold. We prove that the quasi-isomorphism class of the $Frob_\infty^0$-bialgebra structure on $H^*(M)$ induced by the open TFT on $Ω^*(M)$ is a homotopy invariant of the manifold. This is a three step process. First, we describe the $Frob_\infty^0$-bialgebra on $H^*(M)$ induced by the partial $Frob_\infty^0$-bialgebra on $Ω^*(M)$. We then describe the $Frob_\infty^0$-bialgebra on $H^*(M)$ induced by the cyclic $C_\infty$-algebra on $H^*(M)$. Finally, we show these two $Frob_\infty^0$-bialgebras are the same. Since the cyclic $C_\infty$-algebra is a homotopy in…
For a finite group $G$, there is a map $RO(G) \to {\rm Pic}(Sp^G)$ from the real representation ring of $G$ to the Picard group of $G$-spectra. This map is not known to be surjective in general, but we prove that when $G$ is cyclic this map is indeed surjective and in that case we describe ${\rm Pic}(Sp^G)$ explicitly. We also show that for an arbitrary finite group $G$ homology and cohomology with coefficients in a cohomological Mackey functor do not see the part of ${\rm Pic}(Sp^G)$ coming from the Picard group of the Burnside ring. Hence these homology and cohomology calculations can be gra…
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.
For a finite p-group G and a bounded below G-spectrum X of finite type mod p, the G-equivariant Segal conjecture for X asserts that the canonical map X^G --> X^{hG} is a p-adic equivalence. Let C_{p^n} be the cyclic group of order p^n. We show that if the C_p Segal conjecture holds for a C_{p^n} spectrum X, as well as for each of its C_{p^e} geometric fixed points for 0 < e < n, then then C_{p^n} Segal conjecture holds for X. Similar results hold for weaker forms of the Segal conjecture, asking only that the canonical map induces an equivalence in sufficiently high degrees, on homotopy groups …
Equivariant complex $K$-theory and the equivariant sphere spectrum are two of the most fundamental equivariant spectra. For an odd $p$-group, we calculate the zeroth homotopy Green functor of the localization of the equivariant sphere spectrum with respect to equivariant complex $K$-theory. Further, we calculate the zeroth homotopy Tambara functor structure in the case of odd cyclic $p$-groups.
We develop the properties of the $n$-th sequential topological complexity $TC_n$, a homotopy invariant introduced by the third author as an extension of Farber's topological model for studying the complexity of motion planning algorithms in robotics. We exhibit close connections of $TC_n(X)$ to the Lusternik-Schnirelmann category of cartesian powers of $X$, to the cup-length of the diagonal embedding $X\hookrightarrow X^n$, and to the ratio between homotopy dimension and connectivity of $X$. We fully compute the numerical value of $TC_n$ for products of spheres, closed 1-connected symplectic m…
Delooping cyclic groups with lens spaces in homotopy type theory
针对同伦类型论中通用群去环化方法消去原则无法直接处理非截断类型的问题，本文通过透镜空间的类型论实现，构造了循环群ℤₘ的胞腔式去环化，推广了m=2情形的相关结果，并给出二面体群的胞腔描述。
We show that the homotopy groups of a connective $E_k$-ring spectrum with an $E_k$-cell attached along a class $α$ in degree $n$ are isomorphic to the homotopy groups of the cofiber of the self-map associated to $α$ through degree $2n$. Using this, we prove that the $2n-1^{st}$ homotopy groups of Ravenel's $X(n)$ spectra are cyclic for all $n$. This further implies that, after localizing at a prime, $X(n+1)$ is homotopically unique as the $E_1$-$X(n)$-algebra with homotopy groups in degree $2n-1$ killed by an $E_1$-cell. Lastly, we prove analogous theorems for a sequence of $E_k$-ring Thom spe…
Making use of Gruson-Raynaud's technique of "platification par eclatement", Kerz and Strunk proved that the negative homotopy K-theory groups of a Noetherian scheme X of Krull dimension d vanish below -d. In this note, making use of noncommutative algebraic geometry, we improve this result in the case of quotient singularities by proving that the negative homotopy K-theory groups vanish below -1. Furthermore, in the case of cyclic quotient singularities, we provide an explicit "upper bound" for the first negative homotopy K-theory group.
A Quillen model structure of local homotopy equivalences
本文在 ind-Banach 空间投影系统的分次复形范畴上构造闭模型结构，其同伦范畴为局部与解析循环同调理论提供了合适的目标。
The framed little 2-discs operad is homotopy equivalent to the Kimura-Stasheff-Voronov cyclic operad of moduli spaces of genus zero stable curves with tangent rays at the marked points and nodes. We show that this cyclic operad is formal, meaning that its chains and its homology (the Batalin-Vilkovisky operad) are quasi-isomorphic cyclic operads. To prove this we introduce a new complex of graphs in which the differential is a combination of edge deletion and contraction, and we show that this complex resolves BV as a cyclic operad.
Minimal Unital Cyclic $C_\infty$-Algebras and the Real and Rational Homotopy Type of Closed Manifolds
利用\(k\)模同伦概念对有限维分次交换代数\(H^*\)的极小\(C_\infty\)-代数增强集进行分层，证明相关等价条件，定义障碍集并建立广义可加性，确定特定闭流形同伦型，给出新定理证明及边界扩展。
On the geometric fixed points of the real topological cyclic homology of $\mathbb{Z}/4$
本文研究了 $\mathbb{Z}/4$ 的实拓扑循环同调的几何固定点的同调群，揭示了其在不同度数下的结构及渐进行为。
We show that the nerve complex of n arcs in the circle is homotopy equivalent to either a point, an odd-dimensional sphere, or a wedge sum of spheres of the same even dimension. Moreover this homotopy type can be computed in time O(n log n). For the particular case of the nerve complex of evenly-spaced arcs of the same length, we determine the dihedral group action on homology, and we relate the complex to a cyclic polytope with n vertices. We give three applications of our knowledge of the homotopy types of nerve complexes of circular arcs. First, we use the connection to cyclic polytopes to …
Algebraic cycles on complex projective space P(V) are known to have beautiful and surprising properties. Therefore, when V carries a real or quaternionic structure, it is natural to ask for the properties of the groups of real or quaternionic algebraic cycles on P(V). In this paper and its sequel the homotopy structure of these cycle groups is determined. They bear a direct relationship to characteristic classes for the classical groups, and functors in K-theory extend directly to these groups. These groups give rise to E-infinity-ring spaces, and the maps extending the K-theory functors are r…
Cyclic brace relation and BV structure on open-closed Hochschild cohomology
本文研究开闭同伦代数的Hochschild上同调，在循环且带单位的条件下，利用循环括号运算的上链级恒等式构造出BV代数结构。
We give a classification of the $p$--local stable homotopy type of $BG$, where $G$ is a finite group, in purely algebraic terms. $BG$ is determined by conjugacy classes of homomorphisms from $p$--groups into $G$. This classification greatly simplifies if $G$ has a normal Sylow $p$--subgroup; the stable homotopy types then depends only on the Weyl group of the Sylow $p$--subgroup. If $G$ is cyclic mod $p$ then $BG$ determines $G$ up to isomorphism. The last class of groups is important because in an appropriate Grothendieck group $BG$ can be written as a unique linear combination of $BH$'s, whe…
The cyclic bar construction and fundamental groups
本文计算了取值于主理想整环的单纯上链代数的0阶Hochschild同调，证明其由自由环的有限型同伦不变量（即基本群上的有限型类函数）构成，为几何构造3维流形中链环不变量及实现Milnor环绕数提供基础。
We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an annular link $L$ a naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov…
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