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From: Graham Ellis <0002319S@bodkin.nuigalway.ie>
Subject: Research Assistantship
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University College Galway ------------------------- -------------------------Forbairt Research Assistantship in Mathematics ---------------------------------------------- ----------------------------------------------( May 1997 )
A three year Research Assistantship is avaiable in the Mathematics Department
at University College Galway, at a salary of #6,500 p.a. plus travel expenses.
There is also the possibility of some paid teaching hours.
The holder will be expected to work towars a PhD on "A computer aided classif-
ication of low-dimensional homotopy types of spaces."
Applicants should have a master's degree or good first degree in mathematics,
and should have (or be prepared to develop) an interest in topology, algebra,
and computer algebra. A more precise description of the research project is
given below.
For further information please contact Graham Ellis at graham.ellis@ucg.ie .
Project Description ------------------- -------------------
The project is part of a general investigation into the algebraic properties of
low-dimensional topological spaces. The investigation is currently aided by a
Forbairt funded research assistant, Aidan McDermott (aidan.mcdermott@ucg.ie),
and a Forbairt funded Digital alphastation. Three of the results to date are:
1. There are at most p^{nd-n-d} homotopy classes of continuous functions S^3
--> SK(G,1) from the 3-sphere to the suspension of the classifying space of a
d-generator group G of prime-power order p^n (see reference [EM] below).
2. A polyhedron X with only finitely many non-trivial homotopy groups, all of
which are finite, can be faithfully represented by a simplicial group whose
group of n-simplices is finite for each n>=0 (see reference [E2]).
3. A "k-type of order m" is defined to be the set of all topological spaces
that are homotopy equivalent to some given connected polyhedron X whose
homotopy groups Pi_i(X) satisfy Pi_i(X)=0 for i>k, and such that the product
of the orders of the groups Pi_i(X) (0<i<=k) is equal to m. (See for instance
[B]). We let H(k,m) denote the number of homotopy k-types of order m.
A group-theoretic result of G.Higman and C.Sims implies that
H(1,p^n) = P^{(2/27)n^3 + O(n^(8/3))} ;
it is shown in [E3] that
H(2,p^n) = p^{(9/512)n^4 + O(n^3)} ,
H(3,p^n) = p^{(32/9375)n^5 + O(n^4)} .
These explicit estimates for H(k,p^n) lead naturally to the problem of listing
all k-types of order p^n for particular values of n and p.
When k=1 this problem is one of classifying groups of prime-power order. The
groups of order p^5 had been listed by the year 1899, thanks to the efforts of
Bagnera, Cayley, Holder, Young and others. In the 1930's P.Hall worked on the
classification, and introduced his "isoclinism theory". In 1964 M.Hall and
J.Senior classified all 267 groups of order 2^6 according to isoclinism class.
The 2328 groups of order 2^7 were classified in 1980 by Rodemich. Using
computer techniques, M.Newman subsequently corrected some inaccuracies in
Rodemich's list. In 1988 E.O'Brien extended Newman's methods to obtain a
classification of the 56092 groups of order 2^8.
It is envisaged that the Research Assistant will use techniques from [P][E1]
to classify k-types of order p^n, for k=2,3 and for particular values of n and
p. The computational aspects of this work will be carried out on the Digital
Alphastation. The work might involve the following phases:
1. Understand the methods used to classify prime-power groups (such as second
cohomology, isoclinism theory, and the computer algebra package GAP).
2. Understand how polyhedra are described by homotopy groups and cohomology
classes. Perform some hand-calculations for low values of k and n.
3. Implement a computer algorithm for calculating the third integral homology
of a finite group. Using a well-known relationship between homology and
cohomology, this could be used to enumerate 2-types of prime-power order. It
might be possible to extend it to 3-types.
4. Develop an isoclinism theory for k-types.
5. Classify 2-types and 3-types of order 2^n, according to isoclinism class,
for all low values of n.
6. All computer algorithms will be included in a GAP module which is currently
under development at Galway.
References
[B] H.J.Baues, "Homotopy types", in the handbook of Algebraic Topology, ed.
I.M.James, Elsevier (Amsterdam 1995), 1-72.
[E1] G.Ellis, "Homology of 2-types", J. London Math. Soc. (2) 46 (1992), 1-27.
[E2] G.Ellis, "Spaces with finitely many non-trivial homotopy groups all of
which are finite", Topology Vol. 36 (1997), 501-504.
[E3] G.Ellis, "Enumerating prime-power homotopy types, and the homology of a
group extension", submitted to Topology.
[EM] G.Ellis & A.McDermott, "Tensor products of prime-power groups", J. Pure
Applied Algebra (to appear)
[P] M.Postnikov, "Investigations in homotopy theory of continuous mappings", American Math. Soc. Translations: 7 (1957), 1-134; 11 (1959), 115-153. ------------------------------------------------------------------------- Goetz.Pfeiffer@ucg.ie http://schmidt.ucg.ie/~goetz/ University College Galway, Ireland. phone ++353-91-750353