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Homotopical intersection theory
, 2009
"... 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 disjuncti ..."
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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 [HQ], but our proofs are fundamentally different. Contents
POINCARÉ DUALITY AND PERIODICITY
, 707
"... Abstract. We construct periodic families of Poincaré spaces. This gives a partial solution to a question posed by Hodgson in the proceedings of the 1982 Northwestern homotopy theory conference. In producing these families, we formulate a recognition principle for Poincaré duality in the case of fini ..."
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Abstract. We construct periodic families of Poincaré spaces. This gives a partial solution to a question posed by Hodgson in the proceedings of the 1982 Northwestern homotopy theory conference. In producing these families, we formulate a recognition principle for Poincaré duality in the case of finite complexes having one top cell that splits of after a single suspension. We also explain how a Zequivariant version of our constructions yield a new description of the fourfold periodicity appearing in high dimensional knot cobordism. 1.
Toral and exponential stabilization for homotopy spherical spaceforms
"... Abstract The AtiyahSinger equivariant signature formula implies that the products of isometrically inequivalent classical spherical space forms with the circle are not homeomorphic, and in fact the same conclusion holds if the circle is replaced by a torus of arbitrary dimension. These results are ..."
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Abstract The AtiyahSinger equivariant signature formula implies that the products of isometrically inequivalent classical spherical space forms with the circle are not homeomorphic, and in fact the same conclusion holds if the circle is replaced by a torus of arbitrary dimension. These results are important in the study of group actions on manifolds. Algebraic Ktheory yields standard classes of counterexamples for topological and smooth analogs of spherical spaceforms. The results of this paper characterize pairs of nonhomeomorphic topological spherical space forms whose products with a given torus of arbitrary dimension are homeomorphic, and the main result is that the known counterexamples are the only ones that exist. In particular, this and basic results in lower algebraic Ktheory show that if such products are homeomorphic, then the products are already homeomorphic if one uses a 3dimensional torus. Sharper results are established for important special cases such as fake lens spaces. The methods are basically surgerytheoretic with some input from homotopy theory. One consequence is the existence of new infinite families of manifolds in all dimensions greater than three such that the squares of the manifolds are homeomorphic although the manifolds themselves are not. Analogous results are obtained in the smooth category.
POINCARÉ EMBEDDINGS OF SPHERES
"... Abstract. Given a 1connected Poincaré duality space M of dimension 2p, with p> 2, we give criteria for deciding when homotopy classes S p − → M are represented by framed Poincaré embedded pspheres. 1. ..."
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Abstract. Given a 1connected Poincaré duality space M of dimension 2p, with p> 2, we give criteria for deciding when homotopy classes S p − → M are represented by framed Poincaré embedded pspheres. 1.
THE DUALIZING SPECTRUM, II.
, 2007
"... Abstract. To an inclusion H ⊂ G of topological groups, we associate a spectrum DH⊂G, which coincides with the dualizing spectrum DG of [K4] when H = G. We also introduce a fibered spectrum analogue. The main application is to give a purely homotopy theoretic construction of Poincaré embeddings in st ..."
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Abstract. To an inclusion H ⊂ G of topological groups, we associate a spectrum DH⊂G, which coincides with the dualizing spectrum DG of [K4] when H = G. We also introduce a fibered spectrum analogue. The main application is to give a purely homotopy theoretic construction of Poincaré embeddings in stable codimension. 1.
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"... such that (1.1) is a homotopy pushout, (P,∂T) and (C,∂T) are Poincaré duality pairs 1 in dimension n, and the map i is (n − m − 1)connected. The motivating example of a Poincaré embedding arises when W is a closed orientable PLmanifold of dimension n and f: P ֒ → W is a piecewise linear embedding ..."
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such that (1.1) is a homotopy pushout, (P,∂T) and (C,∂T) are Poincaré duality pairs 1 in dimension n, and the map i is (n − m − 1)connected. The motivating example of a Poincaré embedding arises when W is a closed orientable PLmanifold of dimension n and f: P ֒ → W is a piecewise linear embedding of a compact polyhedron P in W. Alternatively we can also take f to be a smooth embedding between smooth compact manifolds. Then f(P) admits a regular neighborhood, that is a codimension 0 compact submanifold T ⊂ W that deformation retracts to P (see [26, page 33].) Let C: = W � T be the closure of the complement of T in W. Then C and T are both compact manifolds of dimension n with a common boundary ∂T = ∂C and W = T ∪∂T C. The composition of the inclusion ∂T ֒ → T with the retraction T ≃ → P gives a map i: ∂T → P and we obtain the pushout (1.1). If the polyhedron P is of dimension m, then a general position argument implies that the map i is (n −m−1)connected. Of course C has the homotopy type of the complement W � f(P).
NONFIBERWISE SIMPLY CONNECTED POINCARÉ SURGERY
"... ABSTRACT. Klein showed that a metastable Poincaré embedding with complement compresses to an embedding iff an invariant vanishes. Klein conjectured that is Poincaré dual to the Hopf invariant of the unstable normal invariant, for a 1connected middle dimensional embedding. We prove this and deduce P ..."
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ABSTRACT. Klein showed that a metastable Poincaré embedding with complement compresses to an embedding iff an invariant vanishes. Klein conjectured that is Poincaré dual to the Hopf invariant of the unstable normal invariant, for a 1connected middle dimensional embedding. We prove this and deduce Poincaré surgery in the simply connected case. 1.
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"... Cyclic group actions on manifolds from deformations of rational homotopy types. ..."
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Cyclic group actions on manifolds from deformations of rational homotopy types.