0.1 Higher dimensional, generalized van Kampen theorems (HD-GVKT)

There are several generalizations of the original van Kampen theorem, such as its extension to crossed complexes, its extension in categorical form in terms of colimits, and its generalization to higher dimensions, i.e., its extension to 2-groupoids, 2-categories and double groupoids [3].

With this HDA-GVKT approach one obtains comparatively quickly not only classical results such as the Brouwer degree and the relative Hurewicz theorem, but also non–commutative results on second relative homotopy groups, as well as higher dimensional results involving the action of, and also presentations of, the fundamental group. For example, the fundamental crossed complex ΠX_∗ of the skeletal filtration of a CW–complex X is a useful generalization of the usual cellular chains of the universal cover of X. It also gives a replacement for singular chains by taking X to be the geometric realization of a singular complex of a space. Non-Abelian higher homotopy (and homology) results in higher dimensional algebra (HDA) were proven by Ronald Brown that generalize the original van Kampen’s theorem for fundamental groups (ordinary homotopy, [10]) to fundamental groupoids ([1]) double groupoids, and higher homotopy ([7]); please see also Ronald Brown’s presentation of the original van Kampen’s theorem at PlanetMath.org [8].

Related research areas are: algebraic topology, higher dimensional algebra (HDA) , higher dimensional homotopy, non-Abelian homology theory, supercategories, axiomatic theory of supercategories, n-categories, lextensive categories, topoi/toposes, double groupoids, omega-groupoids, crossed complexes of groupoids, double categories, double algebroids, categorical ontology, axiomatic foundations of Mathematics, and so on.

Its potential for applications in quantum algebraic topology (QAT), and especially in Non-Abelian Quantum Algebraic Topology (NAQAT) related to QFT, HQFT, TQFT, quantum gravity and supergravity (quantum field) theories has also been recently pointed out and explored ([9, 11, 13]).

0.2 Generalized van Kampen theorem (GvKT)

Consideration of a set of base points leads next to the following theorem for the fundamental groupoid.

0.2.1 The van Kampen theorem for the fundamental groupoid, π₁(X,X₀), [1]

Let the space X be the union of open sets U,V with intersection W, and let X₀ be a subset of X meeting each path component of U,V,W. Then:

• (C) (connectivity) X₀ meets each path component of X, and
• (I) (isomorphism) the diagram of groupoid morphisms induced by inclusions:

is a pushout of groupoids

0.2.2 Remarks

When extended to the context of double groupoids this theorem leads to a higher dimensional generalization of the Van Kampen theorem, the HD-GVKT, [3].

Note that this theorem is a generalization of an analogous Van Kampen theorem for the fundamental group, [1, 10]. From this theorem, one can compute a particular fundamental group π₁(X,x₀) using combinatorial information on the graph of intersections of path components of U,V,W, but for this it is useful to develop the algebra of groupoids. Notice two special features of this result:

• (i) The computation of the invariant one wants to obtain, the fundamental group, is obtained from the computation of a larger structure, and so part of the work is to give methods for computing the smaller structure from the larger one. This usually involves non canonical choices, such as that of a maximal tree in a connected graph. The work on applying groupoids to groups gives many examples of such methods [6, 7].
• (ii) The fact that the computation can be done at all is surprising in two ways: (a) The fundamental group is computed precisely, even though the information for it uses input in two dimensions, namely 0 and 1. This is contrary to the experience in homological algebra and algebraic topology, where the interaction of several dimensions involves exact sequences or spectral sequences, which give information only up to extension, and (b) the result is a non commutative invariant, which is usually even more difficult to compute precisely.

0.2.3 Essential data from ref. [3]

The reason for this success seems to be that the fundamental groupoid π₁(X,X₀) contains information in dimensions 0 and 1, and therefore it can adequately reflect the geometry of the intersections of the path components of U,V,W and the morphisms induced by the inclusions of W in U and V . This fact also suggested the question of whether such methods could be extended successfully to higher dimensions.

References