# TTT104: Sheffield

The University of Sheffield.

Supported by the London Mathematical Society.

## Date

20 July 2017

## Speakers

### Dae-Woong Lee (Chonbuk, Korea)

On the Steenrod homology, co-Hopf structures, SNT and the Brown-Peterson spectra

On the Steenrod homology, co-Hopf structures, SNT and the Brown-Peterson spectra

Abstract: In this talk, we describe some fundamental results about the strong (co)homology groups of inverse systems and phantom groups.

We also describe the set of comultiplications on a wedge of finite number of spheres. We are primarily interested in the size of this set and properties of the comultiplications such as associativity and commutativity. Our methods involve Whitehead products in wedges of spheres and the Hopf-Hilton invariants.

We apply our results to specific examples and determine the number of comultiplications, associative comultiplications and commutative comultiplications in these cases. This is a part of a joint work with Martin A Arkowitz.

By using the rational homotopy theory, we give an answer to the question on the same n-type structure based on the infinite complex projective space raised by C A McGibbon and J M Moller in 1990, and describe the generalized SNT conjectures of CW-complexes. Especially, we show that the set of all the same n-types of the suspension of the Eilenberg-MacLane spaces is the one element set consisting of a single homotopy type.

Finally, we consider the homotopy structure of the real-oriented Brown-Peterson spectrum and the local cohomology of the basic block and the negative block of the homotopy of BPR<3>, and their Anderson duals. This is a joint work in progress with J P C Greenlees.

### Dean Barber (Sheffield)

A combinatorial model for the Fulton-Macpherson operads

A combinatorial model for the Fulton-Macpherson operads

Abstract: The Fulton-Macpherson operads are built from compactified configuration spaces that form manifolds with corners and are one way to realise EN operads. It has been shown that the Fulton-Macpherson operads are cofibrant by claiming the existence of isomorphisms between them and their cofibrant replacements. It would be satisfying if we could write down these isomorphisms explicitly.

In this talk, we introduce a combinatorial model for the Fulton-Macpherson operads which is built from objects called chained linear preorders. These models have an easy to see spine embedded within the spaces, which we believe could be used to create the structured collarings that are integral to the aforementioned isomorphisms.

### Andy Tonks (Leicester)

Tilings, trees, DG2As and B∞-algebras

Tilings, trees, DG2As and B∞-algebras

Abstract: Recall that a B∞-structure may be expressed as a DG bialgebra structure on a cofree coassociative algebra, or as a multibrace algebra with a compatible A∞-structure. Loday and Ronco showed that the primitive part of a cofree Hopf algebra is a multibrace algebra, constructed its universal enveloping 2-associative algebra, and gave descriptions of the free multibrace and 2-associative algebras in terms of trees.

In joint work with Gálvez-Carrillo and Ronco we extend these results to B∞ and differential graded 2-associative algebras, and describe the free objects in terms of guillotine partitions in d=3 dimensions.