algebraic models of homotopy types, crossed modules and the free loop space

algebraic models of homotopy types, crossed modules and the free loop space

[Ronnie Brown]

There has been discussion on directions for higher category theory and
relations with homotopy theory, and I explained that what I have been
trying to do might be better described as `higher dimensional group
theory'. As an example I mention the following:

In March,  Niranjan Ramachandram wrote to me as follows:
-----------------------------------------------

Permit me to ask a question about the free loop space LX = Maps (S1, X)
of a nice connected space X. The path components of LX correspond to
conjugacy classes in the fundamental group of X.

Is there a good place where the fundamental group of the various path
components of LX are calculated? I am particularly interested in the
components corresponding to non-trivial conjugacy classes of the
fundamental group of X.

Perhaps the more natural question is how to describe the fundamental
(and higher homotopy) groupoid of LX in terms of the homotopy invariants
of X. Is there such an elegant description?

----------------------------------------------------

This led me to write an answer to say that if X is the classifying space
of a crossed module of groups, M \to P,  then a crossed module of
groupoids L(M \to P) can be written down explicitly and which describes
completely the homotopy 2-type of L(M \to P); this is rather better than
just describing the fundamental groupoid, or even just the various first
and second homotopy groups! The paper has been through several revisions
and corrections, and the current version is available on
arxiv.org/abs/1003.5617 . I hope the next version will include some
computer calculations by a colleague!

The result is a special case of the description of the weak homotopy
type of (BC)^Y when C is a crossed complex and Y is a CW-complex, in
terms of
CRS(\Pi Y_*,C), using the internal hom for crossed complexes, and \Pi Y*
is the fundamental crossed complex of Y with its skeletal filtration,
which involves relative homotopy groups, and so relies on a 1991 paper
by myself and Higgins on `the classifying space of a crossed complex'.

It is interesting to tease out for which circumstances various proposed
models of homotopy types are `useful', which again depends on one's aims.

The above also generalises the case X=BG is the classifying space of a
group, discussed in my 1987 groupoid survey (Bull LMS) for (BG)^Y for a
general Y.

Ronnie Brown


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