The loop group and the cobar construction

Kathryn Hess Bellwald
2010
Journal paper

We prove that for any 1-reduced simplicial set X, Adams' cobar construction, on the normalised chain complex of X is naturally a strong deformation retract of the normalised chains CGX on the Kan loop group GX, opening up the possibility of applying the tools of homological algebra to transfering perturbations of algebraic structure from the latter to the former. In order to prove our theorem, we extend the definition of the cobar construction and actually obtain the existence of such a strong deformation retract for all 0-reduced simplicial sets.

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Related concepts (4)

Homological algebra

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of .

Algebraic structure

In mathematics, an algebraic structure consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures.

Loop group

In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. In its most general form a loop group is a group of continuous mappings from a manifold M to a topological group G. More specifically, let M = S1, the circle in the complex plane, and let LG denote the space of continuous maps S1 → G, i.e. equipped with the compact-open topology. An element of LG is called a loop in G. Pointwise multiplication of such loops gives LG the structure of a topological group.