Truncated symmetric products and configuration spaces

Bödigheimer, Carl-Friedrich; Cohen, Fred; Milgram, R. James

doi:10.1007/bf02572399

Cited by 32 publications

(29 citation statements)

References 22 publications

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“…One knows that these configuration spaces have isomorphic cohomologies [3], stable homotopy types ( [2] and Cohen and Taylor, private communication), and have homotopy equivalent loop spaces [7,22]. But the homotopy invariance of the configuration spaces themselves is not yet fully understood.…”

Section: Introductionmentioning

confidence: 99%

The homotopy invariance of the string topology loop product and string bracket

Cohen

Klein

Sullivan

2008

Journal of Topology

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Let M n be a closed, oriented, n-manifold, and LM its free loop space. In [Chas and Sullivan, 'String topology', Ann. of Math., to appear] a commutative algebra structure in homology, H * (LM ), and a Lie algebra structure in equivariant homology H S 1 * (LM ), were defined. In this paper, we prove that these structures are homotopy invariants in the following sense. Let f : M1 → M2 be a homotopy equivalence of closed, oriented n-manifolds. Then the induced equivalence, Lf : LM1 → LM2 induces a ring isomorphism in homology, and an isomorphism of Lie algebras in equivariant homology. The analogous statement also holds true for any generalized homology theory h * that supports an orientation of the Mi .

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Section: Introductionmentioning

confidence: 99%

The homotopy invariance of the string topology loop product and string bracket

Cohen

Klein

Sullivan

2008

Journal of Topology

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“…For odd-dimensional orientable manifolds M, the cohomology groups of B(M, n) with rational or modulo p coefficients have been described explicitly by Bödigheimer, Cohen and Milgram [4]. For even-dimensional manifolds the computations are much more difficult.…”

Section: Introductionmentioning

confidence: 99%

On the Cohomology of Configuration Spaces on Surfaces

Napolitano

2003

J. London Math. Soc.

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The integral cohomology rings of the configuration spaces of n-tuples of distinct points on arbitrary surfaces (not necessarily orientable, not necessarily compact and possibly with boundary) are studied. It is shown that for punctured surfaces the cohomology rings stabilize as the number of points tends to infinity, similarly to the case of configuration spaces on the plane studied by Arnold, and the Goryunov splitting formula relating the cohomology groups of configuration spaces on the plane and punctured plane to arbitrary punctured surfaces is generalized. Moreover, on the basis of explicit cellular decompositions generalizing the construction of Fuchs and Vainshtein, the first cohomology groups for surfaces of low genus are given.

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“…One can "read off' the homology of the braid groups, certain groups which are related to mapping class groups, or the homology of unordered configuration spaces directly from the homology of the iterated loop space of a sphere. Similar remarks apply to some ordered configuration spaces and explain the connection between their cohomology, a free Lie algebra, and Whitehead products in classical homotopy theory [BCT,BCM,Cl,C2,C3]. Thus a fruitful point of view is that various function spaces provide a coherent and uniform picture of the types of invariants encountered in this book and in earlier work on these subjects.…”

Section: A Partial Synthesismentioning

confidence: 60%

Book Review: Complements of discriminants of smooth maps\/}: {\it Topology and applications

Cohen¹

1994

Bull. Amer. Math. Soc.

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Truncated symmetric products and configuration spaces

Cited by 32 publications

References 22 publications

The homotopy invariance of the string topology loop product and string bracket

The homotopy invariance of the string topology loop product and string bracket

On the Cohomology of Configuration Spaces on Surfaces

Book Review: Complements of discriminants of smooth maps\/}: {\it Topology and applications

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