An introduction to the homotopy groups of spheres Setting up the Adams spectral sequence The classical Adams spectral sequence $BP$-theory and the Adams-Novikov spectral sequence The chromatic spectral sequence Morava stabilizer algebras Computing stable homotopy groups with the Adams-Novikov spectral sequence Hopf algebras and Hopf algebroids Formal group laws Tables of homotopy groups of spheres Bibliography Index

Complex Cobordism and Stable Homotopy Groups of Spheres

Introduction Polytopes Topology and combinatorics of simplicial complexes Commutative and homological algebra of simplicial complexes Cubical complexes Toric and quasitoric manifolds Moment-angle complexes Cohomology of moment-angle complexes and combinatorics of triangulated manifolds Cohomology rings of subspace arrangement complements Bibliography Index.

/pdf/torus-actions-and-their-applications-in-topology-and-36j92o51jp.pdf

Torus Actions and their Applications in Topology and Combinatorics

Starting with a choice of a gauge group in four dimensions, there is often freedom in the choice of magnetic and dyonic line operators. Different consistent choices of these operators correspond to distinct physical theories, with the same correlation functions of local operators in R 4 . In some cases these choices are permuted by shifting the θ-angle by 2π. In other cases they are labeled by new discrete θ-like parameters. Using this understanding we gain new insight into the dynamics of four-dimensional gauge theories and their phases. The existence of these distinct theories clarifies a number of issues in electric/magnetic dualities of supersymmetric gauge theories, both for the conformal N = 4 theories and for the low-energy dualities of N = 1 theories.

/pdf/reading-between-the-lines-of-four-dimensional-gauge-theories-18jugfx83j.pdf

Reading between the lines of four-dimensional gauge theories

Omar Antoĺın Camarena pointed out a gap in the proofs in [BT&G, Bro06] of a condition for the Phragmen–Brouwer Property not to hold; this note gives the correction in terms of a result on a pushout of groupoids, and some additional background.

https://gendocs.ru/docs/16/15764/conv_1/file1.pdf

Topology and Groupoids

https://hal.archives-ouvertes.fr/hal-00126037/document

On nearly parallel G2-structures

where n = dim M and 6» = ith Betti number of M ( = dim of Hi(M; Q)). Thus the geometric property of M having a nonzero vector field is expressed in terms of the algebraic invariant xM. We will discuss extensions of this idea to vector ^-fields, fields of ^-planes, and foliations of manifolds. All manifolds considered will be connected, smooth and without boundary; all maps will be continuous. For background information on manifolds and vector fields see [30], [34] and [67].

https://projecteuclid.org/download/pdf_1/euclid.bams/1183530621

Vector fields on manifolds

Complete solutions to a family of cubic Diophantine equations

Postnikov Invariants and Higher Order Cohomology Operations

: Contents: Introduction; Principal Fiber Spaces; The classical method of decomposing a fibration; Oriented sphere bundles; Killing homotopy groups; An illustration; and Computing Postnikov invariants.

Emery Thomas

Papers

Vector fields on manifolds

Complete solutions to a family of cubic Diophantine equations

Postnikov Invariants and Higher Order Cohomology Operations

Seminar on fiber spaces

Complex Structures on Real Vector Bundles