Research: Original Scientific Discoveries

Professionally, I am a pure mathematician, working in the Topology of High-Dimensional Manifolds (curriculum vitæ).
I co-organized two research conferences: Nil Phenomena in Topology (2007) and High Dimensional Topology (2012).
I co-organized the advanced summer school, Surgery and the Classification of Manifolds (2016), and also taught Lectures 02 | 07 | 08 | 11.

PUBLICATIONS (original work, peer-review by anonymous expert, selected by an editor)

Immediately after each publication number on the left, you will find the 2010 Journal Quality Ranking of the Australian Research Council:

A* = one of the best in its field or subfield (top 5%)   |   A = very high quality (next 15%)   |   B = solid, though not outstanding (next 30%)   |  
C = does not meet the criteria for the higher tiers (bottom 50%)   |   U = ranking not available because the journal is too new

00:MISC  doctoral dissertation — On connected sums of real projective spaces Indiana University Bloomington, 209 pages, 2006 [ book | review ]

MISC  report — The homotopy type of G/PL and the characteristic variety theorem (with T Macko), Oberwolfach Rep., 3(3): 2212–2214, 2006 [ report ]

01:A*  Manifolds homotopy equivalent to Pn # Pn (with J Brookman and JF Davis), Math. Annalen, 338(4): 947–962, 2007 [ survey | article | review ]

02:A  On smoothable surgery for 4-manifolds Algebr. Geom. Topol., 7: 2117–2140, 2007 [ article | review ]

03:B  On fibering and splitting of 5-manifolds over the circle Topology Appl., 156(2): 284–299, 2008 [ article | review ]

04:A  Reduction of UNil for finite groups with normal abelian Sylow 2-subgroup J. Pure Appl. Algebra, 213(3): 279–298, 2009 [ article | review ]

05:B  Calculation of UNil for the cyclic group of order two Forum Math., 22(2): 221–239, 2010 [ article | review ]

06:B  Dihedral manifold approximate fibrations over the circle (with CB Hughes), Geom. Dedicata, 148(1): 205–243, 2010 [ article | review ]

07:A  Algebraic K-theory over the infinite dihedral group: an algebraic approach (with JF Davis and A Ranicki), Algebr. Geom. Topol., 11(4): 2391–2436, 2011 [ announce | article | review | survey | lecture ]

08:B  Homotopy invariance of 4-manifold decompositions: connected sums Topology Appl., 159(16): 3432–3444, 2012 [ article | review ]

09:A*  Topological rigidity and H1-negative involutions on tori (with F Connolly and JF Davis), Geom. Topol., 18(3): 1719–1768, 2014 [ article | review ]

10:A*  Topological rigidity and actions on contractible manifolds with discrete singular set (with F Connolly and JF Davis), Trans. Amer. Math. Soc., B2: 113–133, 2015 [ announce | article | review | lecture ]

11:B  Cancellation for 4-manifolds with virtually abelian fundamental group Topology Appl., 220: 14–30, 2017 [ article | review | lecture ]

12:A  Free transformations of S1×Sn of square-free odd period Indiana Univ. Math. J., 66(5): 1453–1482, 2017 [ article | review | lecture ]

13:B  Countable approximation of topological G-manifolds: compact Lie groups G Topology Appl., 235: 14–21, 2018 [ article | review ]

14:B  Stable existence of incompressible 3-manifolds in 4-manifolds (with G Smith), Topology Appl., 262: 50–63, 2019 [ article | review | lecture A B C ]

15:B  Countable approximation of topological G-manifolds, II: linear Lie groups G J. Topol. Anal., 14(2) : 557–565, 2022 [ article | review | lecture ]

16:B  Countable approximation of topological G-manifolds, III: arbitrary Lie groups G New York J. Math., 27: 1554–1579, 2021 [ article | review ]

17:B  Cardinal-indexed classifying spaces for families of subgroups of any topological group Topology Appl., 311: #107965, 2022 [ article | review ]

SELECTION OF OPEN RESEARCH PROJECTS (private manuscript)

‣   Noncommutative localization in algebraic L-theory, II: rings with preduality — investigation of [Ranicki 2006]'s speculations and his student [Palmer 2015]'s dissertation reactivated (9 pages @ 29 Mar 2022)

‣   Corrigendum #2 to: Orbit spaces and unions of equivariant absolute neighborhood extensors — thorough correction of the analytic G-polytopes introduced in [§4 Antonyan 2005] (with S Antonyan), agreed on 15 Oct 2020

‣   On recognition of absolute G-extensors, I: finite dimensions — equivariant generalization of [Kodama 1957]/[Dugundji 1958] as organized in [V:7.1 Hu 1965] outlining viability (3 pages @ 29 Sep 2020)

‣   On recognition of absolute G-extensors, II: countable dimensions — equivariant generalization of [Haver 1973] (strongly countable dimensional) and [Gresham 1980] (Property C) (with FD Ancel), deferred until Part I is complete

‣   The 16 homotopy types of manifolds double-covered by T2×S3 — continuation of [Jahren–Kwasik 2011] and of my student [S Arnold 2020]'s dissertation (with G Smith), post-dissertation topic

‣   Finitely many G-orbit types of compact sets in cohomology-manifolds for compact groups G — generalization of [Floyd–Bredon 1960 Borel seminar] (with LR Taylor), preliminary discussion (05 Oct 2019)

‣   Free transformations of Tm×Sn of square-free odd period — continuation of [Khan 2017b] (with T Macko), suspended (3 pages @ 26 Dec 2017)

‣   Contemplation of the module structures on Nil and UNil — grappling with infinite generation in [Farrell 1979] and [Connolly–Davis 2004] suspended (9 pages @ 05 Sep 2016)

‣   Isovariant approximation of G-maps with large gaps for finite groups G — TOP rehash correcting fatal mistake of DIFF [Schultz 2006]; preceded by [Straus 1972]'s dissertation and [Browder 1987]'s announcement (with F Connolly), suspended (since May 2014)

‣   On smoothable surgery for 4-manifolds, II: fundamental groups with 2-torsion — continuation of [Khan 2007] (with D Crowley), suspended (6 pages @ 17 Jul 2011)

‣   Calculation of UNil for cyclotomic and quadratic number rings — continuation of [Banagl–Ranicki 2006] and [Khan 2006]'s dissertation; development from [Weiss 1985 I] suspended (17 pages @ 22 Jun 2008)

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