Dr Wajid Mannan

Research

Research Interests:

I am working with Prof. Michael Farber on a stochastic approach to the Whitehead conjecture.  He has existing results concerning the probability that a randomly generated 2-complex will provide a counterexample to the Whitehead conjecture.  We are attempting to determine if these results are compatible with the existence of a counterexample.

In 2003 Prof. Michael Farber introduced the notion of topological complexity.  This measures the number of rules needed to navigate the moduli space of configurations of a robotic system.  Here a rule continuously assigns to every ordered pair of points a path going from the first point to the second.  We are interested in extending this work to more groups (that is classifying spaces of groups) and spaces.

 

In the past my research has mostly centered around low dimensional cell complexes, particularly in relation to Wall's D2 conjecture.  This asks if a finite 3-complex with no cohomology in dimension 3 is necessarily homotopy equivalent to a finite 2-complex.  This question involves the study of homotopy modules over the fundamental group, representation theory of groups over the integers and group presentations.  As such it is closely tied to questions such as the Eilenberg-Ganea conjecture, the Wiegold conjecture, the relation gap problem, the Kervaire conjecture and the Andrews-Curtis conjecture.

This research led to work on the K-theory of cyclotomic fields, and the Poincare duality of 5-dimensional manifolds.  I have also worked on the Koszul duality of differential graded algebras and the closed model category structure of categories of differential graded coalgebras and (curved) Lie algebras.

 

Publications

[13] W.H. Mannan ; Duality in the homology of 5-manifolds : Homology, Homotopy and Applications Vol. 19 (2017), No. 1, pp. 171-179

http://dx.doi.org/10.4310/HHA.2017.v19.n1.a9

 

 

[12]} W.H. Mannan ; Explicit generators for the relation module in the example of Gruenberg--Linnell : Mathematical Proceedings of the Cambridge Philosophical Society Vol. 161 (2016), Issue 02, pp. 199-202

http://dx.doi.org/10.1017/S0305004116000189

 

[11] J. Chuang, A. Lazarev and W.H. Mannan ; Cocommutative coalgebras: homotopy theory and Koszul duality : Homology, Homotopy and Applications Vol. 18 (2016), No. 2, pp. 303-336

http://dx.doi.org/10.4310/HHA.2016.v18.n2.a17

 

[10]} J. Chuang, A. Lazarev and W.H. Mannan ; Koszul--Morita Duality :  Journal of Noncommutative Geometry Vol. 10 (2016), Issue 4, pp. 1541-1557

http://www.ems-ph.org/journals/show_abstract.php?issn=1661-6952&vol=10&iss=4&rank=8

 

[9] Z. Choo, W.H. Mannan, R. Garcia-Sanchez  and V.P. Snaith ; Computing Borel's regulator :  Forum Mathematicum Vol. 27 (2015), Issue 1, pp. 131-177

http://www.degruyter.com/view/j/forum.2015.27.issue-1/forum-2012-0064/forum-2012-0064.xml?format=INT

 

[8] W.H. Mannan and Seamus O'Shea ; Minimal algebraic complexes over D_4n :  Algebr. Geom. Topol. 13 (2013), Issue 6, pp. 3287-3304

http://msp.org/agt/2013/13-6/p10.xhtml

 

[7]} W.H.Mannan ; A commutative version of the group ring : Journal of Algebra Vol. 379 (2013), pp. 113-143

http://dx.doi.org/10.1016/j.jalgebra.2012.12.026

 

[6]} W.H. Mannan ; Quillen's Plus Construction and the D(2) problem : Algebr. Geom. Topol. 9 (2009), Issue 3, pp. 1399-1411

http://msp.org/agt/2009/9-3/p06.xhtml

 

[5] W.H. Mannan ; Realizing algebraic 2-complexes by cell complexes : Mathematical Proceedings of the Cambridge Philosophical Society Vol. 146 (2009), Issue 03, pp. 671-673

http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=5370832&fulltextType=RA&fileId=S0305004108002107

 

[4]W.H. Mannan ; Periodic Cohomology : Homology, Homotopy and Applications Vol. 10 (2008), No. 2, pp. 135-137

http://intlpress.com/hha/v10/n2/a6/pdf

 

[3] W.H. Mannan ; The third homotopy module of a 2-complex} : Bulletin of the London Mathematical Society 40 (2008), Issue 4, pp. 664-674

http://blms.oxfordjournals.org/cgi/reprint/40/4/664.pdf

 

[2] W.H. Mannan ; Homotopy types of truncated projective resolutions : Homology, Homotopy and Applications Vol. 9 (2007), No. 2, pp. 445-449

http://intlpress.com/hha/v9/n2/a16/pdf

 

[1] W.H. Mannan} ; The D(2) property for D_8 : Algebr. Geom. Topol. 7 (2007), Issue 1, pp. 517-528

http://msp.org/agt/2007/7-1/p22.xhtml