Abstract not found

Let f1,f2: M − → N be two (continuous) maps between smooth connected manifolds M and N without boundary, of strictly positive dimensions m and n, resp., M being compact. We are interested in making the coincidence locus C(f1,f2): = {x ∈ M | f1(x) = f2(x)} as small (or simple in some sense) as possible after possibly deforming f1 and f2 by a homotopy. Question. How large is the minimum number of coincidence components MCC(f1,f2): = min{#π0(C(f ′ 1,f ′ 2)) | f ′ 1 ∼ f1,f ′ 2 ∼ f2}? In particular, when does this number vanish, i.e. when can f1 and f2 be deformed away from one another? This is a very natural generalization of one of the central problems of classical fixed point theory (where M = N and f2 = identity map): determine the minimum number of fixed points among all maps in a given homotopy class (see [Br] and [BGZ], proposition 1.5). Note, however, that in higher codimensions m − n> 0 the coincidence locus is generically a closed (m−n)-manifold so that it makes more sense to count pathcomponents rather than points. Also the methods of (first order, singular) (co)homology will no longer be strong enough to capture the subtle geometry of coincidence manifolds. In this lecture I will use the language of normal bordism theory (and a nonstabilized version thereof) to define and study lower bounds N(f1,f2) (and N #(f1,f2)) for MCC(f1,f2). After performing an approximation we may assume that the map (f1,f2) : M → N × N is smooth and transverse to the diagonal ∆ = {(y,y) ∈ N × N | y ∈ N}. Then the coincidence locus C = C(f1,f2) = (f1,f2) −1 (∆) is a closed smooth (m − n)-dimensional manifold, equipped with i) maps

Given Poincar'e spaces M and X , we study the possibility of compressing embeddings of M \ThetaI in X \ThetaI down to embeddings of M in X . This results in a new approach to embedding in the metastable range both in the smooth and Poincar'e duality categories.

Abstract. We give a new approach to intersection theory. Our “cycles ” are closed manifolds mapping into compact manifolds and our “intersections ” are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn [H-Q], but our proofs are fundamentally different. Contents

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

Abstract. We obtain multirelative connectivity statements about spaces of smooth embeddings, deducing these from analogous results about spaces of Poincaré embeddings that were established in [GK1]. 1.

In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs.f1;f2 / of maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism theory as our main tool. This leads to estimates of the minimum numbers MCC.f1;f2 / (and MC.f1;f2/, resp.) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to.f1;f2/. Furthermore, we deduce finiteness conditions for MC.f1;f2/. As an application we compute both minimum numbers explicitly in various concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E.f1;f2 / into path components. Its higher dimensional topology captures further crucial geometric coincidence data. In the setting of homotopy groups the resulting invariants are closely related to certain Hopf–Ganea homomorphisms which turn out to yield finiteness obstructions for MC. 55M20, 55Q25, 55S35, 57R90; 55N22, 55P35, 55Q40 1

Abstract not found

We review a cochain-free treatment of the classical van Kampen obstruction ϑ to embeddability of an n-polyhedron 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 odd-dimensional analogue of ϑ is a complete obstruction to linkless embeddability (=“intrinsic unlinking”) of the given n-polyhedron in R2n+1. • A “blown up ” 1-parameter 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 integer-valued 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, non-embeddable 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 “k-co-connected and locally k-co-connected ” polyhedra, which we embed in R2n−k for k < n−3 extending the Penrose–Whitehead–Zeeman theorem.

Let M be an oriented manifold and let N be a set consisting of oriented closed manifolds of possibly different dimensions. Roughly speaking, the space GN,M of N-garlands in M is the space of mappings into M of singular manifolds obtained by gluing manifolds from N at some marked points. In our previous work with Rudyak we introduced a rich algebra structure on the bordism group Ω∗(GN,M). In this work we introduce the operations ⋆ and [·, ·] on the tensor product of Q and a certain bordism group ̂Ω∗(GN,M). For N consisting of odd-dimensional manifolds, these operations make ̂Ω∗(GN,M) ⊗ Q into a graded Poisson algebra (Gerstenhaber-like algebra). For N consisting of even-dimensional manifolds, ⋆ satisfies a graded Leibniz rule with respect to [·, ·], but [·, ·] does not satisfy a graded Jacobi identity. The mod 2-analogue of [·, ·] for one-element sets N was previously constructed in our preprint with Rudyak. For N = {S 1} and a surface F 2, the subalgebra ̂ Ω0(G {S 1},F 2) ⊗ Q of our algebra is related to the Goldman-Turaev algebra of loops on a surface and to the Andersen-Mattes-Reshetikhin Poisson algebra of chord-diagrams. As an application, our Lie bracket allows one to compute the minimal number of intersection points of loops in two given homotopy classes ̂ δ1, ̂ δ2 of free loops on F 2, provided that ̂ δ1, ̂ δ2 do not contain powers of the same element of π1(F 2).