Introduction to Axiomatic Set Theory. 2nd ed. 2 OXTOBY. Measure and Category. 2nd ed. 3 SCHAEFER. Topological Vector Spaces. 2nd ed. 4 HILTON/STAMMBACH. A Course in Homological Algebra. 2nd ed. 5 MAC LANE. Categories for the Working Mathematician. 2nd ed. 6 HUGHES/PIPER. Projective Planes. 8 TAKEUTI/ZARING. Axiomatic Set Theory. 9 HUMPHREYS. Introduction to Lie Algebras and Representation Theory. 10 COHEN. A Course in Simple Homotopy Theory. 11 CONWAY. Functions of One Complex Variable I. 2nd ed. 12 BEALS. Advanced Mathematical Analysis. 13 ANDERSON/FULLER. Rings and Categories of Modules. 2nd ed. 14 GOLUBITSKY/GUILLEMIN. Stable Mappings and Their Singularities.

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ScienceFoundation.1 G has type F n if there is a K(G, 1)-complex with finite n-skeleton. All groups have type F 0 , F 1 is "finitely generated", F 2 is "finitely presented", etc.

Abstract. Thompson's group F is the group of all increasing dyadic PL homeomorphisms of the closed unit interval. We compute † m .F / and † m .F I Z/, the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F , and show thatAs an application, we show that, for every m, F has subgroups of type F m 1 which are not of type FP m (thus certainly not of type F m ).Mathematics Subject Classification (2010). 20J05, 20F65, 55U10.

Abstract. The Product Conjecture for the homological Bieri-Neumann-Strebel-Renz invariants is proved over a field. Under certain hypotheses the Product Conjecture is shown to also hold over Z, even though D. Schütz has recently shown that the Conjecture is false in general over Z. Our version over Z is applied in a joint paper with D. Kochloukova [5] to show that for all n Thompson's group F contains subgroups of type F n which are not of type FP nC1 . Mathematics Subject Classification (2010). 20E06, 20F65, 55U25.