University of Regina Topology Seminar

Toda brackets are a type of higher homotopy operation. Like Massey products, they are not always defined, and their value is indeterminate. Nevertheless, they play an important role in algebraic topology and related fields. Toda originally constructed them as a tool for computing homotopy groups of spheres. Adams later showed that they can be used to calculate differentials in spectral sequences.

After reviewing the construction and properties of the classical Toda bracket, we shall describe a higher order version, there are two ways to do that. We will provide a diagrammatic description for the system we need to define the higher order Toda brackets, then we will use that to give an alternative definition using the homotopy cofiber.

Operads are a tool to encode algebraic structure. They were introduced in the 1960s to describe the structure found in loop spaces. They have since found applications in topology, algebra, and mathematical physics. This talk will be an introduction to operads and algebras over them. We will look at examples, including the associative, commutative, and Lie operads.

A loop space is a good example of an $A_{\mathrm{\infty }}$ space as its multiplication is not strictly associative but only associative up to coherent homotopy. In this seminar I will talk about the Stasheff associahedra, $A_{\mathrm{\infty }}$ spaces and the recognition principle for loop spaces.

Like an $A_{\mathrm{\infty }}$ space having a multiplication associative up to coherent homotopy, an $E_{\mathrm{\infty }}$ space has a multiplication which is commutative up to coherent homotopy. Examples include iterated loop spaces. In this seminar I will talk about the little n-cubes operad, iterated loop spaces and the recognition principle.

Many infinite loop spaces can be expressed in terms of the linear isometries operad. In this seminar I will talk about its construction as an $E_{\mathrm{\infty }}$ operad and its properties, in particular, how we can recognize infinite loop spaces using this operad and some applications of this result.

Using the linear isometries operad, I will sketch the construction of L-spectra and S-modules, a model for stable homotopy theory with a well-behaved smash product. One salient feature is that S-algebras and commutative S-algebras model $A_{\mathrm{\infty }}$ ring spectra and $E_{\mathrm{\infty }}$ ring spectra, respectively.

Homotopy invariance is an important property not satisfied by all operads. In our talk, we give counterexamples and discuss some examples involving $A_{\mathrm{\infty }}$ operads in topological spaces and $A_{\mathrm{\infty }}$ algebras in chain complexes.

Topological Hochschild homology (THH) is an important variant for ring spectra. It is built as a geometric realization of a cyclic bar construction. It is endowed with an action of circle. This is because it is a geometric realization of a cyclic object. The simplex category factors through Connes’ category $\mathrm{\Lambda }$. Similarly, real topological Hochschild homology (THR) for ring spectra with anti-involution is endowed with a O(2)-action. Here instead of the cyclic category $\mathrm{\Lambda }$, we use the dihedral category $\mathrm{\Xi }$.

From work in progress with Gabe Angelini-Knoll and Mona Merling, I present a generalization of $\mathrm{\Lambda }$ and $\mathrm{\Xi }$ called crossed simplicial groups, introduced by Fiedorowicz and Loday. To each crossed simplical group G, I define THG, an equivariant analogue of THH. Its input is a ring spectrum with a twisted group action. THG is an algebraic invariant endowed with different action and cyclotomic structure, and generalizes THH and THR.

We look at model categories on the category of operads, on the category algebras over a fixed operad and on the category of modules over a fixed algebra over a fixed operad. We discuss how these can be useful.

Modular operads are particular, heavily-structured operads which show up in many areas of physics. The story of these operads winds its way through Feynman diagrams, quantum field theory, and string theory.

In this talk we shall motivate the construction of modular operads by considering their application to the theory of moduli spaces of Riemann surfaces. Along the way, we shall encounter ways in which physics has inspired (or created) solutions to problems involving Riemann surfaces.

If an algebra $A$ is Koszul, certain subcategories of the derived categories of graded $A$-modules and $A^{!}$-modules form an equivalence. In this talk, I will discuss what it means for an algebra to be Koszul, the Koszul dual of a quadratic algebra, and will focus on some examples (the symmetric algebra and exterior algebra being Koszul dual to each other).