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Fixed point theory and trace for bicategories
, 2007
"... The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point inde ..."
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Cited by 9 (7 self)
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The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point index that give a converse to the Lefschetz fixed point theorem. An important part of this theorem is the identification of these different invariants. We define a generalization of the trace in symmetric monoidal categories to a trace in bicategories with shadows. We show the invariants used in the converse of the Lefschetz fixed point theorem are examples of this trace and that the functoriality of the trace provides some of the necessary identifications. The methods used here do not use simplicial techniques and so generalize readily to other contexts. iii Contents
The van Kampen obstruction and its relatives
, 2009
"... We review a cochainfree treatment of the classical van Kampen obstruction ϑ to embeddability of an npolyhedron into R2n and consider several analogues and generalizations of ϑ, including an extraordinary lift of ϑ which in the manifold case has been studied by J.P. Dax. The following results ar ..."
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Cited by 5 (0 self)
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We review a cochainfree treatment of the classical van Kampen obstruction ϑ to embeddability of an npolyhedron into R2n and consider several analogues and generalizations of ϑ, including an extraordinary lift of ϑ which in the manifold case has been studied by J.P. Dax. The following results are obtained. • The mod2 reduction of ϑ is incomplete, which answers a question of Sarkaria. • An odddimensional analogue of ϑ is a complete obstruction to linkless embeddability (=“intrinsic unlinking”) of the given npolyhedron in R2n+1. • A “blown up ” 1parameter version of ϑ is a universal type 1 invariant of singular knots, i.e. knots in R3 with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integervalued type 1 invariant of singular knots admits an integral arrow diagram ( = Polyak–Viro) formula. • Settling a problem of Yashchenko in the metastable range, we obtain that every PL manifold N, nonembeddable in a given Rm, m ≥ 3(n+1), contains a subset X 2 such that no map N → Rm sends X and N \ X to disjoint sets. • We elaborate on McCrory’s analysis of the Zeeman spectral sequence to geometrically characterize “kcoconnected and locally kcoconnected ” polyhedra, which we embed in R2n−k for k < n−3 extending the Penrose–Whitehead–Zeeman theorem.
RELATIVE FIXED POINT THEORY
, 906
"... The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define relative Lefschetz numbers and Reidemeister traces using traces in bicategories with shadows. We use the fun ..."
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Cited by 4 (4 self)
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The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define relative Lefschetz numbers and Reidemeister traces using traces in bicategories with shadows. We use the functoriality of this trace to identify different forms of these invariants and to prove a relative Lefschetz fixed point theorem and its converse.
HOMOTOPICAL INTERSECTION THEORY, III: MULTIRELATIVE INTERSECTION PROBLEMS
"... Abstract. This paper extends some results of Hatcher and Quinn [HQ] beyond the metastable range. We give a bordism theoretic obstruction to deforming a map f: P → N between manifolds simultaneously off of a collection of pairwise disjoint submanifolds Q1,..., Qj ⊂ N. In a certain range of dimensions ..."
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Abstract. This paper extends some results of Hatcher and Quinn [HQ] beyond the metastable range. We give a bordism theoretic obstruction to deforming a map f: P → N between manifolds simultaneously off of a collection of pairwise disjoint submanifolds Q1,..., Qj ⊂ N. In a certain range of dimensions, our obstruction is the entire story.
The geometric Hopf invariant and double points
, 2010
"... The geometric Hopf invariant of a stable map F is a stable Z/2equivariant map h(F) such that the stable Z/2equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehrmap F of an immersi ..."
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The geometric Hopf invariant of a stable map F is a stable Z/2equivariant map h(F) such that the stable Z/2equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehrmap F of an immersion f: M m � N n in terms of the double point set of f. We interpret the SmaleHirschHaefliger regular homotopy classification of immersions f in the metastable dimension range 3m < 2n−1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.