相关论文: Multiplicative properties of Atiyah duality

We prove that Atiyah duality holds in the $\infty$-category of non-$\mathbb A^1$-invariant motivic spectra over arbitrary derived schemes: every smooth projective scheme is dualizable with dual given by the Thom spectrum of its negative…

Let $\mathrm{Emb}(S^1,M)$ be the space of smooth embeddings from the circle to a closed manifold $M$ of dimension $\geq 4$. We study a cosimplicial model of $\mathrm{Emb}(S^1,M)$ in stable categories, using a spectral version of…

It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions, Symm. We offer the cohomology of the loop space of the suspension of the infinite complex projective space as…

We give a simple universal property of the multiplicative structure on the Thom spectrum of an $n$-fold loop map, obtained as a special case of a characterization of the algebra structure on the colimit of a lax $\mathcal{O}$-monoidal…

We show that, for a finite spectrum $X$, Spanier-Whitehead duality induces an isomorphism between the cohomological and homological Atiyah-Hirzebruch spectral sequences. As an application, it follows that Poincar\'e duality for a Poincar\'e…

For M a closed, connected, oriented manifold, we obtain the Batalin-Vilkovisky (BV) algebra of its string topology through homotopy-theoretic constructions on its based loop space. In particular, we show that the Hochschild cohomology of…

Chas and Sullivan recently defined an intersection product on the homology $H_*(LM)$ of the space of smooth loops in a closed, oriented manifold $M$. In this paper we will use the homotopy theoretic realization of this product described by…

We define a notion of "Frobenius pair", which is a mild generalization of the notion of Frobenius object in a monoidal category. We then show that Atiyah duality for smooth manifolds can be encapsulated in the statement that a certain…

We study the cohomology ring of the complement $\mathcal{M}(\mathcal{A})$ of a manifold arrangement $\mathcal{A}$ in a smooth manifold $M$ without boundary. We first give the concept of monoidal cosheaf on a locally geometric poset…

We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f: X->BF, where BF denotes the classifying space for stable spherical fibrations. To do this, we consider symmetric monoidal models of…

Let $M$ be a closed, oriented manifold of dimension $d$. Let $LM$ be the space of smooth loops in $M$. Chas and Sullivan recently defined a product on the homology $H_*(LM)$ of degree $-d$. They then investigated other structure that this…

We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an…

We give a description of the factorization homology and $E_n$ topological Hochschild cohomology of Thom spectra arising from $n$-fold loop maps $f: A \to BO$, where $A = \Omega^n X$ is an $n$-fold loop space. We describe the factorization…

Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as…

We show that the Chas-Sullivan loop product, a combination of the Pontrjagin product on the fiber and intersection product on the base, makes sense on the total space homology of any fiberwise monoid E over a closed oriented manifold M.…

We construct an S^1-equivariant prospectrum that models the Atiyah dual of a free loop space of a manifold. By applying a suitably completed S^1-equivariant K-theory to the Atiyah dual, we show how to recover the Witten genus of the…

We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a 1-connected closed manifold M. We prove that the loop homology of M is isomorphic to the…

The loop product is an operation in string topology. Cohen and Jones gave a homotopy theoretic realization of the loop product as a classical ring spectrum $LM^{-TM}$ for a manifold $M$. Using this, they presented a proof of the statement…

Let $Q$ denote MacLane's $Q$-construction, and $\otimes$ denote the smash product of spectra. In this paper we construct an equivalence $Q(R)\simeq \mathbb Z\otimes R$ in the category of $A_\infty$ ring spectra for any ring $R$, thus…

We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of…