Details of Grant 

EPSRC Reference:	EP/D033144/1
Title:	The atomic retract containing the bottom cell of loop suspensions: its homology and homotopy exponent
Principal Investigator:	Grbic, Professor J
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Researcher Co-Investigators:
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Department:	Mathematical Sciences
Organisation:	University of Aberdeen
Scheme:	Mathematics Small Grant PreFEC
Starts:	12 August 2005	Ends:	11 March 2006	Value (£):	9,850
EPSRC Research Topic Classifications:	Algebra & Geometry
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
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Summary on Grant Application Form
It is standard in mathematics to investigate an object by breaking it into smaller pieces which are simpler, investigate the piece individually, and then reassemble that information to gain insight into the original object. In this proposal we intend to use such an approach to investigate geometric objects. An elementary example is a filled-in square. It is the product of two intervals. The interval is simpler to understand than square, so if we know some property if the interval and how it behaves when taking a product, we get a new property of the square. A slightly more complicated example is the torus (doughnut), which is the product of two circles. The circle is easier to deal with than the torus, so once we know information about the circle and the behaviour of the product, we know something about the torus. For more complicated geometrical objects, it is not easy to recognise when the object decomposes as a product, or if a decomposition exists, to identify the factors. General methods have been developed which give decompositions of certain kinds of geometrical objects which are of a fundamental nature. Examples include n-dimensional spheres, certain manifolds which are important in physics. The aim of this proposal is to use decomposition methods to gain insight into particular types of properties of these fundamental geometrical objects. For example, if there is a multiplication on the geometrical object, we wish to examine to what extent that multiplication is preserved by the decomposition.In all of this the process is made easier by introducing some flexibility in the geometrical objects and how they decompose. We allow the geometrical objects to be deformed continuously, where we can push, pull, stretch, or squeeze, but not tear or puncture. Fore example, the corners of a square can be smoothed out to give a circle, so we regard the square and the circle as the same geometrical object. On the other hand, the only way to get an annulus out of a solid disc is to puncture the disc, so we regard the annulus and the disc as different geometrical objects. This kind of flexibility by deformation has proved very useful and is the subject of homotopy theory, which is the area of mathematics addressed by the proposal. This idea of flexibility can be partially quantified by algebra. The algebra is more concrete than the geometry, in the sense that it lets us directly to calculate things. One aim of this proposal is to decompose certain geometric objects related to the fundamental ones mentioned above, examine some general properties of the algebra associated to each factor, and then investigate to what extent these properties are preserved by the decomposition. In this way we can discover interesting properties of the original geometrical objects.
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