Publications
5 results found
Nicholson J, 2023, Projective modules and the homotopy classification of (G,n)-complexes, Algebraic and Geometric Topology, ISSN: 1472-2739
Kasprowski D, Nicholson J, Ruppik B, 2022, Homotopy classification of 4-manifolds whose fundamental group is dihedral, Algebraic and Geometric Topology, Vol: 22, Pages: 2915-2949, ISSN: 1472-2739
We show that the homotopy type of a finite oriented Poincar\'{e} 4-complex isdetermined by its quadratic 2-type provided its fundamental group is finite andhas a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreckand Bauer, this applies in the case of smooth oriented 4-manifolds whosefundamental group is a finite subgroup of SO(3). An important class of examplesare elliptic surfaces with finite fundamental group.
NICHOLSON J, 2021, A cancellation theorem for modules over integral group rings, Mathematical Proceedings of the Cambridge Philosophical Society, Vol: 171, Pages: 317-327, ISSN: 0305-0041
<jats:title>Abstract</jats:title><jats:p>A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups <jats:italic>G</jats:italic> for which the integral group ring ℤ<jats:italic>G</jats:italic> has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that ℤ<jats:italic>G</jats:italic> has SFC provided at most one copy of the quaternions ℍ occurs in the Wedderburn decomposition of the real group ring ℝ<jats:italic>G</jats:italic>. This generalises the Eichler condition in the case of integral group rings.</jats:p>
Nicholson J, 2021, On CW-complexes over groups with periodic cohomology, Transactions of the American Mathematical Society, Vol: 374, Pages: 6531-6557, ISSN: 0002-9947
If $G$ has $4$-periodic cohomology, then D2 complexes over $G$ are determinedup to polarised homotopy by their Euler characteristic if and only if $G$ hasat most two one-dimensional quaternionic representations. We use this to solveWall's D2 problem for several infinite families of non-abelian groups and, inthese cases, also show that any finite Poincar\'{e} $3$-complex $X$ with$\pi_1(X)=G$ admits a cell structure with a single $3$-cell. The proof involvescancellation theorems for $\mathbb{Z} G$ modules where $G$ has periodiccohomology.
Bokor I, Crowley D, Friedl S, et al., 2021, Connected Sum Decompositions of High-Dimensional Manifolds, 2019-20 MATRIX Annals, Pages: 5-30, ISSN: 2523-3041
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