Related papers: An Additivity theorem for cobordism categories

A notion of vector field cobordism for oriented manifolds was defined by B\"okstedt and Svane. We extend this notion to define complex section cobordism for almost complex manifolds. We then determine the complex section cobordism groups…

We extend the Bass-Quillen Fundamental Theorem of Algebraic K-theory to the Waldhausen K-theory of connective ${\bf S}$-algebras. The same technique used in this extension also yields two localization theorems for the relative K-theory of a…

We prove a twisting theorem for nodal classes in permutation-equivariant quantum $K$-theory, and combine it with existing theorems of Givental to obtain a twisting result for general characteristic classes of the virtual tangent bundle.…

In this paper we introduce the following new ingredients: (1) rework on part of the Lagrangian surgery theory; (2) constructions of Lagrangian cobordisms on product symplectic manifolds; (3) extending Biran-Cornea Lagrangian cobordism…

Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence. We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological…

In this paper, we prove the K- and L-theoretical Isomorphism Conjecture for Baumslag-Solitar groups with coefficients in an additive category.

Cobordism categories are known to be compact closed. They can therefore be used to define non-degenerate models of multiplicative linear logic by combining the Int construction with double glueing. In this work we detail such construction…

In this paper we study cobordism categories consisting of manifolds which are endowed with geometric structure. Examples of such geometric structures include symplectic structures, flat connections on principal bundles, and complex…

This paper provides an informal sketch of a proof of the Baez-Dolan cobordism hypothesis, which provides a classification for extended topological quantum field theories.

We give a new proof of an index theorem for fiber bundles of compact topological manifolds due to Dwyer, Weiss, and Williams, which asserts that the parametrized $A$-theory characteristic of such a fiber bundle factors canonically through…

This paper reformulates Goodwillie calculus of $\infty$-categories including non-presentable $\infty$-categories. In the case of presentable $\infty$-categories our definition is equivalent to Heuts's~\cite{Heuts2018} work. As an…

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the…

We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside $\mathcal{R}^+ (S^d)$ if $d \geq 6$. To achieve…

We introduce and study a $K$-theory of twisted bundles for associative algebras $A(\mathfrak g)$ of formal series with an infinite-Lie algebra coefficients over arbitrary compact topological spaces. Fibers of such bundles are given by…

We consider Hilsum's notion of bordism as an equivalence relation on unbounded $KK$-cycles and study the equivalence classes. Upon fixing two $C^*$-algebras, and a $*$-subalgebra dense in the first $C^*$-algebra, a…

We establish formulas for computation of the higher algebraic $K$-groups of the endomorphism rings of objects linked by a morphism in an additive category. Let ${\mathcal C}$ be an additive category, and let $Y\ra X$ be a covariant morphism…

We define Grothendieck-Witt spectra in the setting of Poincar\'e $\infty$-categories and show that they fit into an extension with a K- and an L-theoretic part. As consequences we deduce localisation sequences for Verdier quotients, and…

Quillen's algebraic K-theory is reconstructed via Voevodsky's algebraic cobordism. More precisely, for a ground field k the algebraic cobordism P^1-spectrum MGL of Voevodsky is considered as a commutative P^1-ring spectrum. There is a…

In this short note, for a morphism of Waldhausen categories $f\colon \mathbb{A} = (\mathcal{A} ,w_{\mathbb{A}}) \to \mathbb{B} = (\mathcal{B},w_{\mathbb{B}})$, we will define $\operatorname{Cone} f$ to be a Waldhausen category. There exists…

We prove the exactness of a descent sequence relating the algebraic cobordism groups of a scheme and its envelopes. Analogous sequences for Chow groups and K-theory were previously proved by Gillet.