Twisted topological Hochschild homology of Cn-equivariant spectra was introduced by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell, building on the work Hopkins, Ravenel norms in equivariant homotopy theory. In this paper we introduce tools for computing twisted THH, which apply to computations Thom spectra, Eilenberg-MacLane real bordism spectrum MUR. particular, construct an versio...

A countable CW complexK is quasi-finite (as defined by A.Karasev [30]) if for every finite subcomplex M of K there is a finite subcomplex e(M) containing M such that any map f : A → M , A closed in a separable metric space X satisfying XτK, there is an extension g : X → e(M) of f . Levin’s [36] results imply that none of the Eilenberg-MacLane spaces K(G, 2) is quasi-finite if G = 0. In this pap...

We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. By using the Dold-Thom functor it can therefore be obtained from the Khovanov homotopy type constructed by Lipshitz and Sarkar. A Khovanov homotopy type is a way of associating a (stable) space to each link L so th...

We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an Eilenberg-MacLane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an assoc...

The notion of a Garside group was first introduced in a paper of Dehornoy and Paris [14]. Over the past decade they have been used as a tool to better understand the structure of Artin’s braid groups [2] and their generalizations. In general, one can use the Garside structure associated with a Garside group to solve the word and conjugacy problems, as well as create a finite dimensional Eilenbe...

We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism category. The objects of this category are wide subcategories of the module category of a hereditary algebra. A morphism [T ] : A → B is an equivalence class of rigid objects T in the cluster category of A so that B is the right hom-ext perpendicular category of the underlying object |T | ∈ A. Facto...

Definition 1.2. Let n ≥ 1 and let G be an abelian group. The fundamental class of K(G, n) is the cohomology class ιn ∈ H (K(G, n);G) corresponding to idG via the isomorphism H n (K(G, n);G) ∼= HomZ(G,G). More explicitly, let ψ : πnK(G, n) ∼= −→ G be some chosen identification, and let h : πn (K(G, n)) ∼= −→ Hn (K(G, n);Z) denote the Hurewicz morphism, defined by h(α) = α∗(un), where un ∈ Hn(S) ...

Let T (j) be the dual of the jth stable summand of Ω2S3 (at the prime 2) with top class in dimension j. Then it is known that T (j) is a retract of a suspension spectrum, and that the homotopy colimit of a certain sequence T (j) → T (2j) → . . . is an infinite wedge of stable summands of K(V, 1)’s, where V denotes an elementary abelian 2 group. In particular, when one starts with T (1), one get...

Associated with each of the classical cohomology theories in algebra has been a theory relating H (H as classically numbered) to obstructions to non-singular extensions and H with coefficients in a “center” to the non-singular extension theory (see [Eilenberg & MacLane (1947), Hochschild (1947), Hochschild (1954), MacLane (1958), Shukla (1961), Harrison (1962)]). In this paper we carry out the ...