A fake Klein bottle with bubble
We resolve the question of the existence of a finite 2-complex with the same fundamental group and Euler characteristic as a Klein bottle with a bubble, but homotopically distinct to it.
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arXiv subjects
Wajid Mannan
Publications and source records attributed to Wajid Mannan.
Low dimensional algebraic complexes over integral group rings
The realization problem asks: When does an algebraic complex arise, up to homotopy, from a geometric complex? In the case of 2- dimensional algebraic complexes, this is equivalent to the D2 problem, which asks when homological methods can distinguish between 2 and 3 dimensional complexes. We approach the realization problem (and hence the D2 problem) by classifying all possible algebraic 2- complexes and showing that they are realized. We show that if a dihedral group has order 2n, then the algebraic complexes over it are parametrized by their second homology groups, which we refer to as algebraic second homotopy groups. A cancellation theorem of Swan ([11]), then allows us to solve the realization problem for the group D8. Let X be a finite geometric 2- complex. Standard isomorphisms and Schanuel's lemma imply that the stable class of pi_2(X) is determined by pi_1(X). We show how pi_3(X) may be calculated similarly. Specifically,we show that as a module over the fundamental group, pi_3(X) is the symmetric part of the module pi_2(X) otimes pi_2(X). As a consequence, we are able to show that when the order of pi_1(X) is odd, the stable class of pi_3(X) is also determined by pi_1(X). Given a closed, connected, orientable 5- dimensional manifold, with finite fundamental group, we may represent it, up to homotopy equivalence, by an algebraic complex. Poincare duality induces a homotopy equivalence between this algebraic complex and its dual. We consider how similar this homotopy equivalence may be made to the identity, (through appropriate choice of algebraic complex). We show that it can be taken to be the identity on 4 of the 6 terms of the chain complex. However a homological obstruction prevents it from being the identity.
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A commutative version of the group ring
We construct a commutative version of the group ring and show that it allows one to translate questions about the normal generation of groups into questions about the generation of ideals in commutative rings. We demonstrate this with an alternative proof of a result about the normal generation of the free product of two cyclic groups.
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Homotopy types of truncated projective resolutions
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. We show they may be stabilized by projective modules to obtain a pair of complexes of the same homotopy type.
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Cocommutative coalgebras: homotopy theory and Koszul duality
We extend a construction of Hinich to obtain a closed model category structure on all differential graded cocommutative coalgebras over an algebraically closed field of characteristic zero. We further show that the Koszul duality between commutative and Lie algebras extends to a Quillen equivalence between cocommutative coalgebras and formal coproducts of curved Lie algebras.
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Realizing algebraic 2-complexes by cell complexes
The realization theorem asserts that for a finitely presented group G, the D(2) property and the realization property are equivalent as long as G satisfies a certain finiteness condition. We show that the two properties are in fact equivalent for all finitely presented groups.
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Explicit generators for the relation module in the example of Gruenberg-Linnell
Gruenberg and Linnell showed that the standard relation module of a free product of $n$ groups of the form $C_r \times \mathbb{Z}$ could be generated by just $n+1$ generators, raising the possibility of a relation gap. We explicitly give such a set of generators.
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The third homotopy module of a 2-complex
Given a connected 2-complex X with fundamental group G, we show how pi_3(X) may be computed as a module over Z[G]. Further we show that if X is a finite connected 2-complex with G (the fundamental group) finite of odd order, then the stable class of pi_3(X) is determined by G.
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The D(2) property for $D_8$
Wall's D(2) problem asks if a cohomologically 2-dimensional geometric 3-complex is necessarily homotopy equivalent to a geometric 2-complex. We solve part of the problem when the fundamental group is dihedral of order $2^n$, and offer a complete solution for the case where it is $D_8$ the dihedral group of order 8.
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Quillen's Plus Construction and the D(2) problem
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 to the Cayley complex of a finite group presentation. This reduces the D(2) problem to a question about perfect normal subgroups.
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Minimal algebraic complexes over $D_{4n}$
We show that cancellation of free modules holds in the stable class $Ω_3(\mathbb{Z})$ over dihedral groups of order $4n$. In light of a recent result on realizing $k$-invariants for these groups, this completes the proof that all all dihedral groups satisfy the D(2) property.
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Periodic cohomology
We offer a direct proof of an elementary result concerning cohomological periods. As a corollary we show that given a finitely generated stably free resolution of Z over a finite group, two of its modules are free.
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Duality in the homology of 5-manifolds
We show that the homological properties of a 5-manifold M with fundamental group G are encapsulated in a G-invariant stable form on the dual of the third syzygy of Z. In this notation one may express an even stronger version of Poincare duality for M. However we find an obstruction to this duality.
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A Random Graph Growth Model
A growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start of the process the pool contains $N$ virtual vertices and no edges. Each time a vertex is sampled and occupied, the edges linking the vertex to previously occupied vertices are added to the pool of virtual elements. We focus on the edge-counting at times when the graph has $n\leq N$ occupied vertices. Two different Poisson limits are identified for $n\asymp N^{1/3}$ and $N-n\asymp 1$. For the bulk of the process, when $n\asymp N$, the scaled number of edges is shown to fluctuate about a deterministic curve, with fluctuations being of the order of $N^{3/2}$ and approximable by a Gaussian bridge.
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An exotic presentation of Q_28
We introduce a new family of presentations for the quaternion groups and show that for the quaternion group of order 28, one of these presentations has non-standard second homotopy group.
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Koszul-Morita duality
We construct a generalization of Koszul duality in the sense of Keller--Lefèvre for not necessarily augmented algebras. This duality is closely related to classical Morita duality and specializes to it in certain cases.
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Computing Borel's Regulator
We present an infinite series formula based on the Karoubi-Hamida integral, for the universal Borel class evaluated on H_{2n+1}(GL(\mathbb{C})). For a cyclotomic field F we define a canonical set of elements in K_3(F) and present a novel approach (based on a free differential calculus) to constructing them. Indeed, we are able to explicitly construct their images in H_{3}(GL(\mathbb{C})) under the Hurewicz map. Applying our formula to these images yields a value V_1(F), which coincides with the Borel regulator R_1(F) when our set is a basis of K_3(F) modulo torsion. For F= \mathbb{Q}(e^{2πi/3}) a computation of V_1(F) has been made based on our techniques.
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