On the Homotopy Type of Certain Two-Dimensional Complexes

Cockcroft, W. H.; Swan, Richard G.

doi:10.1112/plms/s3-11.1.193

Cited by 45 publications

(20 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition to those already mentioned, there are the many other works of Swan [22], [23], [24], [26], and the pioneering contributions of Cockroft and Swan [6], and Dyer and Sieradski [7], [8].…”

Section: Introductionmentioning

confidence: 99%

Stable modules and Wall's D(2)-problem

Johnson¹

2003

Commun Math Helv

View full text Add to dashboard Cite

Abstract. The D(2)-problem is to determine whether for a three-dimensional complex X, the vanishing of 3-dimensional cohomology, in all coefficients, is enough to guarantee that X is homotopically two-dimensional. We show that for finite complexes with finite fundamental group, a positive solution to the D(2)-problem is obtained precisely when all stably free algebraic 2-complexes are geometrically realizable.The proof makes very strong use of techniques which apply to finite fundamental groups but not more generally; in particular, Yoneda's Theorem that, for finite groups, group cohomology is representable by stable modules of finite type, and also the Swan-Jacobinski Cancellation Theorem for such stable modules.

show abstract

Section: Introductionmentioning

confidence: 99%

Stable modules and Wall's D(2)-problem

Johnson¹

2003

Commun Math Helv

View full text Add to dashboard Cite

show abstract

“…For basic properties of rings with these first three properties we suggest [9]. For additional examples of rings satisfying property (3), see [8] and the references given there. By finding conditions for an embedding of a (non-commutative) ring in a skew field to be possible, P. M. Cohn discovered the theory of free ideal rings, in which properties (1) -(5) above play an important role (see, e.g., [12]).…”

Section: Introductionmentioning

confidence: 99%

Leavitt path algebras having Unbounded Generating Number

Abrams

Nam

Phuc

2017

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

Abstract. We present a result of P. Ara which establishes that the Unbounded Generating Number property is a Morita invariant for unital rings. Using this, we give necessary and sufficient conditions on a graph E so that the Leavitt path algebra associated to E has UGN. We conclude by identifying the graphs for which the Leavitt path algebra is (equivalently) directly finite; stably finite; Hermite; and has cancellation of projectives.Mathematics Subject Classifications: 16S99, 18G05, 05C25

show abstract

“…Let X be any 7r-complex with x(X) > Xn + 1-By a theorem of J. H. C. Whitehead [13], We conclude that their algebraic 2-types (77,7r2, k) differ only by the obstruction invariant k EH3(n, tt2) s Z|".|. But each k E H3(ir, tt2) which is the obstruction invariant for a n-complex must be a generator of Hz(tt, tt2) (see [3]). There are exactly (¿>(l7rl) such generators.…”

mentioning

confidence: 99%