Homological and topological properties of locally indicable groups

manuscripta mathematica - Tập 44 - Trang 71-93 - 1983
James Howie1, Hans Rudolf Schneebeli2
1University of Glasgow, Glasgow, UK
2Wettingen, Switzerland

Tóm tắt

The classes of locally indicable groups, conservative groups andD-groups have each been defined in a different context, and have been studied for various reasons. These three classes are shown to coincide. The corresponding mod p versions of the classes are also shown to coincide, for any prime p. Applications to topology are given. In particular, new light is shed on work of Adams on a problem of Whitehead concerning asphericity in 2-complexes.

Tài liệu tham khảo

ADAMS, J.F.: A new proof of a theorem of Cockcroft. J. London Math. Soc.49, 482–488 (1955). BIERI, R.: Normal subgroups in duality groups and in groups of cohomological dimension 2. J.Pure Appl. Alg.7, 35–51 (1976). BRANDENBURG, J., DYER, M.: On J.H.C. Whitehead's aspherical problem I. Comment. Math. Helv.56, 431–446 (1981). BRANDENBURG, J., DYER, M., STREBEL, R.: On J.H.C. Whitehead's aspherical problem II. To appear in Proc. Symposia Pure Math. (Amer. Math. Soc.). BRODSKII, S.D.: Equations over groups and groups with one defining relator (Russian). Uspehi Mat. Nauk.35, 4, 183 (1980) COHEN, J.M.: Aspherical 2-complexes. J. Pure Appl. Alg.12, 101–110 (1978). GERSTEN, S.M.: Conservative groups, indicability, and a conjecture of Howie. Preprint (1981). HIGMAN, G.: The units of group rings. Proc. London Math. Soc. (2)46, 231–248 (1940). HOWIE, J.: Aspherical and acyclic 2-complexes. J. London Math. Soc. (2)20, 549–558 (1979). HOWIE, J.: On the fundamental group of an almost-acyclic 2-complex. Proc. Edinburgh Math. Soc.24, 119–122 (1981). HOWIE, J.: On pairs of 2-complexes and systems of equations over groups. J. reine angew. Math.324, 165–174 (1981). HOWIE, J.: On locally indicable groups. Math. Z.180, 445–461 (1982). MILNOR, J.: On the 3-dimensional Brieskorn manifolds M(p,q,r). Ann. Math. Studies84, 175–225 (1975). SHORT, H.: Topological methods in group theory; the adjunction problem. Thesis, University of Warwick (1981). STALLINGS, J.: Homology and central series of groups.J. Alg.2, 170–181 (1965). STAMMBACH, U.: Anwendungen der Homologietheorie der Gruppen auf Zentralreihen und auf Invarianten von Präsentierungen. Math. Z.94, 157–177 (1966). STREBEL, R.: Homological methods applied to the derived series of groups. Comment. Math. Helv.49, 302–322 (1974). SWAN, R.G.: Projective modules over finite groups. Bull. Amer. Math. Soc.65, 365–367 (1959). WALDHAUSEN, F.: Algebraic K-theory of generalised free products. Ann. Math.108, 135–256 (1978). WHITEHEAD, J.H.C.: On adding relations to homotopy groups. Ann. Math.42, 409–428 (1941). WHITEHEAD, J.M.: Projective modules and their trace ideals. Comm. Alg.8, 1873–1901 (1980).