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59
Topological Gauge Theories and Group Cohomology
, 1989
"... We show that three dimensional ChernSimons gauge theories with a compact gauge group G (not necessarily connected or simply connected) can be classified by the integer cohomology group H 4 (BG, Z). In a similar way, possible WessZumino interactions of such a group G are classified by H 3 (G, Z). ..."
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Cited by 115 (2 self)
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We show that three dimensional ChernSimons gauge theories with a compact gauge group G (not necessarily connected or simply connected) can be classified by the integer cohomology group H 4 (BG, Z). In a similar way, possible WessZumino interactions of such a group G are classified by H 3 (G, Z). The relation between three dimensional ChernSimons gauge theory and two dimensional sigma models involves a certain natural map from H 4 (BG, Z) to H 3 (G, Z). We generalize this correspondence to topological ‘spin ’ theories, which are defined on three manifolds with spin structure, and are related to what might be called Z2 graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.
The basic gerbe over a compact simple Lie group
"... Abstract. Let G be a compact, simply connected simple Lie group. We give a construction of an equivariant gerbe with connection on G, with equivariant 3curvature representing a generator of H 3 G(G, Z). Technical tools developed in this context include a gluing construction for gerbes and a theory ..."
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Cited by 29 (1 self)
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Abstract. Let G be a compact, simply connected simple Lie group. We give a construction of an equivariant gerbe with connection on G, with equivariant 3curvature representing a generator of H 3 G(G, Z). Technical tools developed in this context include a gluing construction for gerbes and a theory of equivariant bundle gerbes. 1.
Geometry of Deligne Cohomology
, 1996
"... The aim of this paper is to give a geometric interpretation of holomorphic and smooth Deligne cohomology. Before stating the main results we recall the definition and basic properties of Deligne cohomology. Let X be a smooth complex projective variety and let Ωr X be the sheaf of germs of holomorphi ..."
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Cited by 27 (0 self)
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The aim of this paper is to give a geometric interpretation of holomorphic and smooth Deligne cohomology. Before stating the main results we recall the definition and basic properties of Deligne cohomology. Let X be a smooth complex projective variety and let Ωr X be the sheaf of germs of holomorphic rforms on X. The qth Deligne complex of X is the complex of sheaves Z(q)D:
Stabilization for the Automorphisms of free groups with boundaries, II
"... In this followup to our earlier paper [6], we improve substantially the dimension range of the homology stability results for mapping class groups of general 3manifolds, and at the same time bypass a gap in that paper. The new results, stated as Theorems 2.1–2.5, have a stable dimension range of s ..."
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Cited by 13 (7 self)
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In this followup to our earlier paper [6], we improve substantially the dimension range of the homology stability results for mapping class groups of general 3manifolds, and at the same time bypass a gap in that paper. The new results, stated as Theorems 2.1–2.5, have a stable dimension range of slope 2 rather than slope 3. They arise from a new technique vaguely reminiscent of an hprinciple. For the special case of automorphism groups of free groups, when the manifold is a punctured connected sum of S1 × S2 ’s, the new technique gives a significant simplification in the previous proofs of homology stability [3, 4]. This technique can also be used to simplify proofs of other homology stability theorems, such as Harer’s theorem for mapping class groups of surfaces. The gap in [6] occurs in section 4 in the proof of assertions (A) and (B), at the point where there are diagram chasing arguments in two commutative diagrams. In each diagram the groups Gn in the two rows are isomorphic but not identical. If we denote by Gn and G ′ n these two isomorphic groups, the first diagram chase needs the composition Hi(Gn) → Hi(Gn+1) → Hi(Gn+1, G ′ n) to be trivial, which is
Lie Groups and pCompact Groups
, 1998
"... A pcompact group is the homotopical ghost of a compact Lie group; it is the residue that remains after the geometry and algebra have been stripped away. This paper sketches the theory of pcompact groups, with the intention of illustrating the fact that many classical structural properties of compa ..."
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Cited by 11 (1 self)
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A pcompact group is the homotopical ghost of a compact Lie group; it is the residue that remains after the geometry and algebra have been stripped away. This paper sketches the theory of pcompact groups, with the intention of illustrating the fact that many classical structural properties of compact Lie groups depend only on homotopy theoretic considerations.
Tverberg partitions and BorsukUlam theorems
 Paci J. Math
, 1997
"... An Ndimensional real representation E ofa finite group G is said to have the “BorsukUlam Property ” ifany continuous Gmap from the (N + 1)fold join of G (an Ncomplex equipped with the diagonal Gaction) to E has a zero. This happens iff the “Van Kampen characteristic class ” of E is nonzero, so ..."
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Cited by 10 (0 self)
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An Ndimensional real representation E ofa finite group G is said to have the “BorsukUlam Property ” ifany continuous Gmap from the (N + 1)fold join of G (an Ncomplex equipped with the diagonal Gaction) to E has a zero. This happens iff the “Van Kampen characteristic class ” of E is nonzero, so using standard computations one can explicitly characterize representations having the BU property. As an application we obtain the “continuous ” Tverberg theorem for all prime powers q, i.e., that some q disjoint faces ofa (q − 1)(d + 1)dimensional simplex must intersect under any continuous map from it into affine dspace. The “classical” Tverberg, which makes the same assertion for all linear maps, but for all q, is explained in our setup by the fact that any representation E has the analogously defined “linear BU property ” iff it does not contain the trivial representation.
CW simplicial resolutions of spaces, with an application
"... Abstract. We show how a certain type of CW simplicial resolutions of spaces by wedges of spheres may be constructed, and how such resolutions yield an obstruction theory for a given space to be a loop space. 1. ..."
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Cited by 7 (5 self)
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Abstract. We show how a certain type of CW simplicial resolutions of spaces by wedges of spheres may be constructed, and how such resolutions yield an obstruction theory for a given space to be a loop space. 1.
Yang–Mills theory over surfaces and the AtiyahSegal theorem
, 2008
"... Abstract. In this paper we explain how Morse theory for the YangMills functional can be used to prove an analogue, for surface groups, of the AtiyahSegal theorem. Classically, the AtiyahSegal theorem relates the representation ring R(Γ) of a compact group Γ to the complex Ktheory of the classify ..."
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Cited by 7 (5 self)
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Abstract. In this paper we explain how Morse theory for the YangMills functional can be used to prove an analogue, for surface groups, of the AtiyahSegal theorem. Classically, the AtiyahSegal theorem relates the representation ring R(Γ) of a compact group Γ to the complex Ktheory of the classifying space BΓ. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson’s deformation Ktheory spectrum Kdef(Γ) (the homotopytheoretical analogue of R(Γ)). Our main theorem provides an isomorphism in homotopy K ∗ def (π1Σ) ∼ = K ∗ (Σ) for all compact, aspherical surfaces Σ and all ∗> 0. Combining this result with work of Lawson, we obtain homotopy theoretical information about the stable moduli space of flat connections over surfaces. 1.