  In the max-min allocation problem a set $P$ of players are to be allocated
disjoint subsets of a set $R$ of indivisible resources, such that the minimum
utility among all players is maximized. We study the restricted variant, also
known as the Santa Claus problem, where each resource has an intrinsic positive
value, and each player covets a subset of the resources. Bez\'akov\'a and Dani
showed that this problem is NP-hard to approximate within a factor less than
$2$, consequently a great deal of work has focused on approximate solutions.
The principal approach for obtaining approximation algorithms has been via the
Configuration LP (CLP) of Bansal and Sviridenko. Accordingly, there has been
much interest in bounding the integrality gap of this CLP. The existing
algorithms and integrality gap estimations are all based one way or another on
the combinatorial augmenting tree argument of Haxell for finding perfect
matchings in certain hypergraphs. Our main innovation in this paper is to
introduce the use of topological methods for the restricted max-min allocation
problem, to replace the combinatorial argument. This approach yields
substantial improvements in the integrality gap of the CLP. In particular we
improve the previously best known bound of $3.808$ to $3.534$. We also study
the $(1,\varepsilon)$-restricted version, in which resources can take only two
values, and improve the integrality gap in most cases.
