Escardó and Simpson defined a notion of interval object by a universal property in any category with binary products. The Homotopy Type Theory book defines a higher-inductive notion of reals, and suggests that the interval may satisfy this universal property. We show that this is indeed the case in the category of sets of any universe. We also show that the type of HoTT reals is the least Cauch...

A classic result of Swan states that a finite group G acts freely on a finite homotopy sphere if and only if every abelian subgroup of G is cyclic. Following this result, Benson and Carlson conjectured that a finite group G acts freely on a finite complex with the homotopy type of n spheres if the rank of G is less than or equal to n. Recently, Adem and Smith have shown that every rank two fini...

A b str act . Let Σ be the image of a topological embedding of Sn−2 into Sn. In this paper the homotopy groups of the complement Sn−Σ are studied. In contrast with the situation in the smooth and piecewise linear categories, it is shown that the first nonstandard homotopy group of the complement of such a topological knot can occur in any dimension in the range 1 through n − 2. If the first non...

In this paper we study the homotopy type of Hom (Cm, Cn), where Ck is the cyclic graph with k vertices. We enumerate connected components of Hom (Cm, Cn) and show that each such component is either homeomorphic to a point or homotopy equivalent to S1. Moreover, we prove that Hom (Cm, Ln) is either empty or is homotopy equivalent to the union of two points, where Ln is an n-string, i.e., a tree ...

We set up foundations of representation theory over S, the stable sphere, which is the “initial ring” of stable homotopy theory. In particular, we treat S-Lie algebras and their representations, characters, gln(S)-Verma modules and their duals, HarishChandra pairs and Zuckermann functors. As an application, we construct a Khovanov slk-stable homotopy type with a large prime hypothesis, which is...

In this paper we study the homotopy type of Hom (Cm, Cn), where Ck is the cyclic graph with k vertices. We enumerate connected components of Hom (Cm, Cn) and show that each such component is either homeomorphic to a point or homotopy equivalent to S1. Moreover, we prove that Hom (Cm, Ln) is either empty or is homotopy equivalent to the union of two points, where Ln is an n-string, i.e., a tree ...

The problem of defining Semi-Simplicial Types (SSTs) in Homotopy Type Theory (HoTT) has been recognized as important during the Year of Univalent Foundations at the Institute of Advanced Study [14]. According to the interpretation of HoTT in Quillen model categories [5], SSTs are type-theoretic versions of Reedy fibrant semi-simplicial objects in a model category and simplicial and semi-simplic...

We give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y ) have the same homotopy type. We describe certain group-like actions on map(X, Y ). Our basic results assert that if maps f, g : X → Y are in the same orbit under such an action, then the components of map(X, Y ) that contain f and g have the same homotopy type.

We compute in this note the full homotopy type of the space of symplectic embeddings of the standard ball B(c) ⊂ R (where c = πr is the capacity of the standard ball of radius r) into the 4-dimensional rational symplectic manifold Mμ = (S × S, μω0 ⊕ ω0) where ω0 is the area form on the sphere with total area 1 and μ belongs to the interval (1, 2]. We know, by the work of Lalonde-Pinsonnault [8]...

We compute in this note the full homotopy type of the space of symplectic embeddings of the standard ball B(c) ⊂ R (where c = πr is the capacity of the standard ball of radius r) into the 4-dimensional rational symplectic manifold Mμ = (S 2 × S, μω0 ⊕ ω0) where ω0 is the area form on the sphere with total area 1 and μ belongs to the interval (1, 2]. We know, by the work of Lalonde-Pinsonnault, ...