By Ol'shanskii A. Y.

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Gk−1 gk−1 is at most m − 1 times. Let F (m) EΣn denote the geometric realization of this simplicial set. Smith conjectured the following result, which was later proved by C. Berger. 2. F (m) EΣn has the homotopy type of the configuration space of n-tuples of distinct points in Rmn . This result can be reformulated in the context of Coxeter groups as follows. Let G be a finite Coxeter group generated by a collection of hyperplanes {Hi } in Rn . Then we can define the Smith filtration on E∗ G by counting the number of times a generic point in Rn gets flipped around any one of the generating hyperplanes by the sequence g0 g1−1 , g1 g2−1 , .

In all known examples for the conjecture, the order of the Sylow p-subgroup of U p (π) is an upper bound for the order of torsion in π∗ (Bπp∧ ). There are examples where this bound is sharp. It is known that for any finite group π, the p-torsion in the homology of the loop space Ω(Bπ)∧p is bounded above by the order of the Sylow p-subgroup of Op (π). 2. Some examples are known. A few are given by π = A5 , A6 , A7 , J4 , M11 at p = 2. A few more at the prime 2 are given by those finite simple groups of 2-rank 2 (including M11 ) with the possible exception of U (3, F4 ).

It is a nonnegative real number. If Y → X is a regular covering of a finite CW complex X with group of GUIDO’S BOOK OF CONJECTURES 57 deck transformations Γ, the Euler characteristic of X can be calculated from the L2 -Betti numbers of Y by the formula: (2) χ(X) = (−1)i L2 bi (Y ; Γ). 1. It is usually called the Singer Conjecture. Beno Eckmann also discusses it in his note to you. 3. Suppose M n is an aspherical manifold with fundamental group π and universal cover M n . Then L2 bi (M n ; π) = 0 for all i = n2 .

### A. I. Maltsevs problem on operations on groups by Ol'shanskii A. Y.

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