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% Loop spaces and homotopy operations
% David Blanc
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% April 27, 1995
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We describe two obstruction theories for a given topological space X to be
a loop space, both defined in terms of higher homotopy operations:
First, we explain how an H-space structure on X can be used to
define the action of the primary homotopy operations on the shifted
homotopy groups \pi_{\star-1} X (which are isomorphic to \pi_{\star} Y
if X\simeq\Omega Y). \ This action will behave properly
with respect to composition of operations if X is homotopy-associative,
and will lift to a topological action of the monoid of all maps between
spheres if and only if X is a loop space. The obstructions
to having such a topological action may be stated in terms of the author's
obstruction theories for realizing Pi-algebras and their morphisms.
A more concrete approach, which does not require a given H-space
structure on X, yields the following:
Theorem A: If X is a CW complex such that all Whitehead products vanish in
\pi_{\star} X, then X is homotopy equivalent to a loop space if and only if
a certain collection of higher homotopy operations vanish coherently.
The higher homotopy operations in question depend only on maps
between wedges of spheres, and take value in homotopy groups of
spheres. They are constructed by means of a certain collection of
convex polyhedra which may be of independent interest.