The realization theorem asserts that for a finitely presented group G, the D(2) property and are equivalent as long G satisfies certain finiteness condition. We show two properties in fact all groups.

Given a finite connected 3-complex with cohomological dimension 2, we show it may be constructed up to homotopy by applying the Quillen plus construction Cayley complex of group presentation. This reduces D(2) problem question about perfect normal subgroups.

We extend a construction of Hinich to obtain closed model category structure on all differential graded cocommutative coalgebras over an algebraically field characteristic zero.We further show that the Koszul duality between commutative and Lie algebras extends Quillen equivalence formal coproducts curved algebras.

Wall's D(2) problem asks if a cohomologically 2-dimensional geometric 3-complex is necessarily homotopy equivalent to 2-complex. We solve part of the when fundamental group dihedral order $2^n$, and offer complete solution for case where it $D_8$ 8.

Abstract We present an infinite series formula based on the Karoubi–Hamida integral, for universal Borel class evaluated H 2 n +1 (GL(ℂ)). For a cyclotomic field F we define canonical set of elements in K 3 ( ) and novel approach (based free differential calculus) to constructing them. Indeed, are able explicitly construct their images (GL(ℂ)) under Hurewicz map. Applying our these yields value V 1 ), which coincides with regulator R when is basis modulo torsion. = ℚ( e 2π i /3 computation...

We introduce a new family of presentations for the quaternion groups and show that group order 28, one these has non-standard second homotopy group.

We show that cancellation of free modules holds in the stable class 3 .Z/ over dihedral groups order 4n.In light a recent result on realizing k -invariants for these groups, this completes proof all satisfy D(2) property.

We work over an arbitrary ring R. Given two truncated projective resolutions of equal length for the same module we consider their underlying chain complexes. show they may be stabilized by modules to obtain a pair complexes homotopy type.

We extend a construction of Hinich to obtain closed model category structure on all differential graded cocommutative coalgebras over an algebraically field characteristic zero. further show that the Koszul duality between commutative and Lie algebras extends Quillen equivalence formal coproducts curved algebras.

Abstract We resolve the question of existence a finite 2‐complex with same fundamental group and Euler characteristic as Klein bottle bubble, but homotopically distinct to it.

We construct a generalization of Koszul duality in the sense Keller–Lefèvre for not necessarily augmented algebras. This is closely related to classical Morita and specializes it certain cases.

Abstract Gruenberg and Linnell showed that the standard relation module of a free product n groups form C r × $\mathbb{Z}$ could be generated by just + 1 generators, raising possibility gap. We explicitly give such set generators.

We offer a direct proof of an elementary result concerning cohomological periods.As corollary we show that given finitely generated stably free resolution Z over finite group, two its modules are free.

We construct a commutative version of the group ring and show that it allows one to translate questions about normal generation groups into ideals in rings. demonstrate this with an alternative proof result free product two cyclic groups.

A growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start process contains $N$ no Each time a vertex sampled occupied, edges linking to previously occupied are added elements. We focus on edge-counting at times when has $n\leq N$ vertices. Two different Poisson limits identified for $n\asymp N^{1/3}$ $N-n\asymp 1$. For bulk process, N$, scaled number shown fluctuate about deterministic curve, with...

We resolve the question of existence a finite 2-complex with same fundamental group and Euler characteristic as Klein bottle bubble, but homotopically distinct to it.

Abstract A growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start process, contains no Each time a vertex sampled occupied, edges linking to previously occupied are added elements. We focus on edge‐counting at times when has vertices. Two different Poisson limits identified for . For bulk , scaled number shown fluctuate about deterministic curve, with fluctuations being order approximable Gaussian bridge.

This paper has been merged with arXiv:0908.3765

We show that Wall's D(2) problem, the Realization problem and Relation Gap could all be solved if it shown deficiency of a certain group is, as intuition would suggest, less than -1. Note paper has been withdrawn. A presentation *_p (C_p x C_p)with -1 is given on p35 of: Cynthia Hog-Angeloni, Beitrage zum (einfachen) homotopietyp zweidimensionaler komplexe zu freein produkten und anderen gruppentheoretischen konstruktionen : PhD thesis, Frankfurt/Main 1988

Given a connected 2-complex X with fundamental group G, we show how pi_3(X) may be computed as module over Z[G]. Further that if is finite G (the group) of odd order, then the stable class determined by G.

We show that the homological properties of a 5-manifold M with fundamental group G are encapsulated in G-invariant stable form on dual third syzygy Z. In this notation one may express an even stronger version Poincare duality for M. However we find obstruction to duality.