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Mathematics Preprints 2006


Algebraic Topology



06.01 : BROWN, R.

Groupoids, the Phragmen-Brouwer Property, and the Jordan Curve Theorem

Summary:

We publicise a proof of the Jordan Curve Theorem which relates it to the Phragmen-Brouwer Property, and whose proof uses the van Kampen theorem for the fundamental groupoid on a set of base points.

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Published in:

J. Homotopy Theory and Related Structures 1 (2006) 175-183.

06.02 : BAK, A., BROWN, R., MINIAN, G. & PORTER, T.

Global actions, groupoid atlases, and applications

Summary:

Global actions were introduced by A. Bak to give a combinatorial approach to higher K-theory, in which control is kept of the elementary operations involved through paths and paths of paths. We give a further application, to identities among relations for groups, extend the theory to groupoid atlases, and develop the homotopy theory of the latter structures. This allows for exploration and development of a kind of atlas of local algebraic structures.

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Published in:

J. Homotopy Theory and Related Structures 1 (2006) 101-167.

06.03 : BROWN, R.

Three themes in the work of Charles Ehresmann: Local-to-global; Groupoids; Higher dimensions.

Abstract:

This paper illustrates the themes of the title in terms of:
  • van Kampen type theorems for the fundamental groupoid;
  • holonomy and monodromy groupoids; and
  • higher homotopy groupoids.
Interaction with work of the writer is explored.

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Expansion of an invited talk given to the 7th Conference on the Geometry and Topology of Manifolds:
The Mathematical Legacy of Charles Ehresmann, Bedlewo 8.05.2005-15.05.2005 (Poland).

(See the discussion site on the web.)

Published in:

Proceedings of the 7th Conference on the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann,
Bedlewo 8.05.2005-15.05. (2005), Poland, Banach Centre Publications.

06.17 : MARTINS, J.F. & PORTER, T.

On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups

Abstract:

We give an interpretation of Yetter's Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. We use this interpretation to define a twisting of Yetter's Invariant by cohomology classes of crossed modules, defined as cohomology classes of their classifying spaces, in the form of a state sum invariant. In particular, we obtain an extension of the Dijkgraaf-Witten Invariant of manifolds to categorical groups. The straightforward extension to crossed complexes is also considered.

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Published in:

TAC 18 (2007) 118-150.

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