Dr Wajid Hassan Mannan

Low dimensional topology

Many questions have been resolved in high dimensions but remain open in the more familiar low dimensions.   The most famous example of this was the Poincare conjecture, but many other problems of this type remain unsolved.  Wall's D(2) problem asks if the dimension of a homotopy type of finite cell complexes is the same as its cohomological dimension.  In 1965 Wall showed that this could only fail if the homotopy type was geometrically 3 dimensional but cohomologically 2 dimensional.  Whether or not this can actually happen has gone on to become one of the major questions in low dimensional topology and is closely related to others, such as the realisation problem, and the Andrews--Curtis Conjecture. 

I have worked extensively on Wall's D(2) problem.  In [8, 1] I resolve the problem for spaces with finite dihedral fundamental group.  In [12] I give an explicit description of a previously known potential counter example.  In [5] I use algebraic techniques developed in [2] to complete the proof that Wall's D(2) problem is equivalent to the realisation problem.

In [6] I show that any counterexample to Wall's D(2) problem will have the form X+ for some 2 dimensional cell complex X.  (Here + denotes Quillen's plus construction).  This further relates Wall's D(2) problem to group theoretic questions such as the Relation Gap Problem (which asks if the number of relators required to present a group can exceed the homological lower bound).  

It was commonly believed that spaces whose second homotopy group had strange number theoretic properties, (called exotic pi_2's) might provide a counterexample to Wall's D(2) problem. However in recent work [14] I show that the prototypical example of such spaces, known colloquially as Nancy's Toy, fails to be a counterexample.  This came as a surprise to the community and alters the landscape of the field.

The homotopy groups of cell complexes may be regarded as modules over the fundamental  group.  For 2 dimensional complexes pi_2 may be computed from the attaching maps, explicitly as a module.  Over a fixed fundamental group the modules pi_2(X) for finite cell complexes X are all equivalent up to stabilization by free modules.  In [3] I  compute pi_3(X) explicitly as a module and show that for a finite group of odd order, the modules pi_3(X) are equivalent up to stabilization by free modules.  For finite groups of even order they are only equivalent if you additionally allow stabilization by one other module, as well as free modules.

For a 5 dimensional manifold X, Poincare duality yields a chain homotopy equivalence f between the cellular chain complex C_*(X~), and its dual.  In [13] I use algebraic techniques developed in [2] to identify  $C_(X~)$ with its dual, leading one to ask if one may assume without loss of generality that f is the identity (up to sign).  I find an obstruction to this  and present manifolds where f cannot be the identity up to sign.

Homological algebra

Much of my doctoral studies was concerned with projective resolutions of modules.  In [2] I showed that homotopy types of projective resolutions of the same module are isomorphic up to a certain stabilization.  In [4] I showed that non-free stably free modules cannot occur at certain points in stably free resolutions over finite groups.  

Quantum algebra

I worked with A. Lazarev and J. Chuang under an EPSRC funded project Homological algebra of Feynman graphs.  In [11] we establish a CMC structure on the category of coalgebras.  Further we establish a CMC structure on the category of curved Lie algebras and demonstrate that these are related by a Quillen equivalence.  In [10] we construct a common generalization of derived Koszul duality in the sense of Keller-Lefevre and Positselski and contravariant Morita equivalence.

Number theory

In 2007 Vic Snaith was leading an EPSRC funded project to investigate the K-theory of cyclotomic fields in relation to the Kummer-Vandiver conjecture, by approximating  the Borel regulator.  By the time I arrived they had a working implementation for  approximating the Borel regulator maps, but needed to find explicit elements in the K-theory of these fields to apply the maps to and approximate the Borel regulator.  This is generally known to be a difficult problem.

I was able to resolve the deadlock in the project by applying ideas from my earlier topological work, in particular the idea of a free differential calculus.  My new approach generated much excitement in the number theory community at the time [9].  This cross fertilization of ideas between number theory and topology is also what led to the results in [6], mentioned above.

Group theory

It is in general hard to prove necessary criteria for a set to normally generate a subgroup.  Major conjectures in this area include the Wiegold conjecture, the Kervaire conjecture, and questions regarding group presentations such as the Relation gap problem mentioned above.  These questions have close links to areas of low dimensional topology such as Dehn surgery on knots and (as previously discussed) I have several results concerning their relation to Wall's D(2) problem [5, 6].

The fundamental problem is that we do not have adequate obstructions to a set of elements in a group (or normal subgroup) normally generating the whole group (resp. normal subgroup).  In [7] I address this problem by introducing a functor # from the category of groups to the category of commutative rings.  I further show that questions about the normal generation of groups (or normal subgroups) get translated into questions concerning the generation of ideals in commutative rings.

The power of this approach is demonstrated in [7] with an alternative proof of a result first proved by Boyer in 1988.  This states that the free product of two cyclic groups C_p * C_q, cannot be normally generated by a proper power; w^r, with r>1.  Using the machinery of [7], this reduces to proving the `high school algebra' statement that a polynomial in one variable of positive degree cannot divide 1.

Random graphs

With Prof. Michael Farber and Prof. Alexander Gnedin I investigated random processes for constructing graphs [15].  The novelty of our process is that at each step either an external edge or vertex may be added.  We identified the critical exponent alpha, where with probability tending to 1 an edge emerges after n^alpha steps.

We have also analysed what happens at the critical value, identifying the coefficient on n^\alpha for the expected number of steps till an edge emerges.

We also modelled the growth of the number of vertices and edges using differential equations.