In this note we discuss how the first author came upon the Kervaire invariant question while analyzing the image of the J-homomorphism in the EHP sequence. One of the central projects of algebraic topology is to calculate the homotopy classes of maps between two finite CW complexes. Even in the case of spheres – the smallest non-trivial CW complexes – this project has a long and rich history. L...

For an arbitrary symmetric monoidal∞-category V, we define the factorization homology of V-enriched∞-categories over (possibly stratified) 1-manifolds and study its basic properties. In the case that V is cartesian symmetric monoidal, by considering the circle and its self-covering maps we obtain a notion of unstable topological cyclic homology, which we endow with an unstable cyclotomic trace ...

We discuss the equivalence between a string theory and the two-dimensional Yang-Mills theory with SU(N) gauge group for finite N . We divide the partition function of the finite N Yang-Mills theory into two sectors, and show that one of these can be interpreted as a sum of covering maps from closed string world-sheets to the target space, whose covering number is less than N . This provides an ...

We give a classification of all equivelar polyhedral maps on the torus. In particular, we classify all triangulations and quadrangulations of the torus admitting a vertex transitive automorphism group. These are precisely the ones which are quotients of the regular tessellations {3, 6}, {6, 3} or {4, 4} by a pure translation group. An explicit formula for the number of combinatorial types of eq...

Given a continuous map f : K → M from a 2-dimensional CW complex into a closed surface, the Nielsen root number N f and the minimal number of roots μ f of f satisfy N f ≤ μ f . But, there is a number μC f associated to each Nielsen root class of f, and an important problem is to know when μ f μC f N f . In addition to investigate this problem, we determine a relationship between μ f and μ ̃ f , ...

A matrix model is constructed which describes a chiral version of the large N U(N) gauge theory on a two-dimensional sphere of area A. This theory has three separate phases. The large area phase describes the associated chiral string theory. An exact expression for the free energy in the large area phase is used to derive a remarkably simple formula for the number of topologically inequivalent ...

We study here the spectra of random lifts of graphs. Let G be a finite connected graph, and let the infinite tree T be its universal cover space. If λ1 and ρ are the spectral radii of G and T respectively, then, as shown by Friedman [Fri03], in almost every n-lift H of G, all “new” eigenvalues of H are ≤ O ( λ 1/2 1 ρ 1/2 ) . Here we improve this bound to O ( λ 1/3 1 ρ 2/3 ) . It is conjectured...

1.2. We refer to [LMB] for the basic background on stacks. Let us quickly recall: a stack over a scheme S is a sheaf of groupoids on the faithfully flat topology of S-schemes, where the presheaf requirement that the composition of the restriction maps is the restriction map of the composition is understood in the weak sense. Stacks form a 2-category with fiber products and there is a clear embe...

A branchpoint of a compactum X is a point which is the vertex of a simple triod in X. A surjective map /: X -» Y is said to cover the branchpoints of Y if each branchpoint in Y is the image of some branchpoint in X. If every map in a class % of maps on a class of compacta & covers the branchpoints of its image, then it is said that the branchpoint covering property holds for ff on 0. According ...

Given a vector space V and a non-singular quadratic form Q on V , the orthogonal group O(Q) is the subgroup of GL(V ) that preserves Q. The special orthogonal group SO(Q) is given by O(Q) ∩ SL(V ). If Q is the standard inner product on R then SO(Q) is denoted SO(n). This is a connected, compact Lie group. For n > 2, the group is semi-simple. In the case that n is odd, SO(n) corresponds to the D...