We analyze in homological terms the homotopy fixed point spectrum of a T-equivariant commutative S-algebra R. There is a homological homotopy fixed point spectral sequence with E^2_{s,t} = H^{-s}_{gp}(T; H_t(R; F_p)), converging conditionally to the continuous homology H^c_{s+t}(R^{hT}; F_p) of the homotopy fixed point spectrum. We show that there are Dyer-Lashof operations beta^epsilon Q^i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the $^{2r}-term of the spectral sequence there are 2r other classes in the E^{2r}-term (obtained mostly by Dyer-Lashof operations on x) that are infinite cycles, i.e., survive to the E^infty-term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH(B) of many S-algebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups C of T, and for the Tate- and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic K-theory of commutative S-algebras.
We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.
要用无界链复形来做同调代数，首先需要找到一种构造分解的方法。斯帕莱蒂解决了这个问题的链复形的R-模块截断进一步向左，解决件，并粘回部分决议。我们的目的是给同伦理论解释这个过程中，这可能会扩展到一个相对的设置。我们在任意的阿贝尔范畴A中工作，并固定一类I的“内射对象”。我们证明了Spaldom的构造可以被无界链复形和非正分次复形之间的一对伴随函子捕获。这对伴随函子形成了我们所说的奎伦对，上面的截断、部分分解和粘合过程给出了一种有意义的方法来解决相对设置中的分裂误差项的复形。为了做同伦理论，特别是为了构造一个行为良好的相对导出范畴D（A; I），我们需要更多：分裂误差项必须为零。这是当I是所有内射R-模的类时的情况，但不是一般的，甚至不是诺特环上的某些内射模类。关键性质是阿贝尔范畴的Roos AB 4 *-n公理的相对类似物。文中还讨论了Gorenstein同调代数和纯性等具体例子。
In this addendum, we give a differential form interpretation of the proof of the main theorem of arXiv:1812.02448, which gives lower bounds of the dimensions of $π_k(B\mathrm{Diff}(D^4,\partial))\otimes\mathbb{Q}$ in terms of the dimensions of Kontsevich's graph homology, and explain why it can be extended to arbitrary even dimensions $d\geq 4$. We attempted to make the proof accessible to more readers. Thus we do not assume familiarity with configuration space integrals nor knowledge of finite type invariants. Part of this addendum might be joined to the original article when it will be re-su…
The primary goal of this paper is to find a homotopy theoretic approximation to moduli spaces of holomorphic maps Riemann surfaces into complex projective space. There is a similar treatment of a partial compactification of these moduli spaces of consisting of irreducible stable maps in the sense of Gromov-Witten theory. The arguments follow those from a paper of G. Segal on the topology of the space of rational functions.
Computing Small 1-Homological Models for Commutative Differential Graded Algebras
我们使用同调扰动机械具体的代数范畴[P。同调扰动理论与结合论。Homology，Homotopy and Applications第2卷，n. 5（2000）51-88]给出了计算交换微分分次代数（简称CDGA）的小1-同调模型的微分结构的算法。的过程的复杂性进行了研究，并在Mathematica的计算机软件包中描述用于确定这样的模型。
In Proc Math Sci 129, 70(219), Rakesh Pawar considers and solves a certain extension problem. In this note, we observe that the existence and uniqueness of differential characters (defined as objects which fit into a hexagon diagram) follow directly from Rakesh Pawar's results. This provides an alternate proof of a weaker version of the results of J. Simons and D. Sullivan, Journal of Topology, 2008, vol 1:45-56. Further this approach directly shows that the hexagon diagram uniquely determines the differential K-theory groups upto an isomorphism.
This article provides a definition of a subdifferential for continuous functions based on homological considerations. We show that it satisfies all the requirement for a good notion of subdifferential. Moreover, we prove sublinearity, a Leibniz formula and an approximation result. This work fits in the framework of microlocal analysis of sheaves for C^0 symplectic problems and application to Aubry-Mather theory.
We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in…
This paper deals with the homotopy theory of differential graded operads. We endow the Koszul dual category of curved conilpotent cooperads, where the notion of quasi-isomorphism barely makes sense, with a model category structure Quillen equivalent to that of operads. This allows us to describe the homotopy properties of differential graded operads in a simpler and richer way, using obstruction methods.
Starting from a biased definition of a properad, we describe explicitly algebras over the cobar construction of a properad. Equivalent description in terms of solutions of generalized master equations, which can be interpreted as homological differential operators, are explained from the properadic point of view. This is parallel to the Barannikov's theory for modular operads. In addition to well known IBL-homotopy algebras, the examples include their associative analogues, which we call $IBA$-homotopy algebras, and a combination of the above two.
We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of the arrangement.
This master's thesis contains an introduction to $A_\infty$-algebras and homological perturbation theory. We then discuss the formality of compact Kähler manifolds and present a direct proof of a homotopy transfer principle of $A_\infty$-algebras, also known as Kadeishvili's Theorem.
