## Guido's book of conjectures by Chatterji I.L. (ed.)

By Chatterji I.L. (ed.)

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The contents of this booklet were used in classes given via the writer. the 1st was once a one-semester direction for seniors on the collage of British Columbia; it was once transparent that strong undergraduates have been completely in a position to dealing with easy workforce concept and its software to easy quantum chemical difficulties.

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1. A uniformly discrete bounded geometry metric space X has property A if and only if the uniform Roe algebra Cu∗ (X) is exact. In evidence for the conjecture we offer the following. The conjecture is true for any countable discrete group equipped with its natural coarse structure. It is then an easy exercise to show that the conjecture holds for any metric space which admits a proper co-compact action by a group of isometries. We proved recently 23 that the conjecture also holds if the space is sufficiently group-like in the following sense.

He then shows that on the one hand the colimit has the same homotopy type as the homotopy colimit, and thus the same homotopy type as the nerve of the poset. On the other hand, he shows by a Quillen Theorem A argument that the nerve of the poset has the same homotopy type as F (m) EΣn . Berger’s poset has a natural interpretation in the Coxeter context. However there are certain technical difficulties in carrying out the complete proof. It may also be the case that there are further generalizations possible for more general classes of reflection groups.

Cat 46 GUIDO’S BOOK OF CONJECTURES 20. Nat` alia Castellana, Juan A. Crespo and J´ erˆ ome Scherer Cohomological finiteness conditions: spaces versus H-spaces We wish to ask a very naive and classically flavored question. Consider a finite complex X and an integer n. Does its n-connected cover X n satisfy any cohomological finiteness property? When X is an H-space we have an answer. 1. Let X be a finite H-space and n an integer. Then H ∗ (X n ; Fp ) is finitely generated as an algebra over the Steenrod algebra.