On string topology of three manifolds

Let M be a closed, oriented and smooth manifold of dimension d. Let LM be the space of smooth loops in M. In [String topology, preprint math.GT/9911159] Chas and Sullivan introduced the loop product, a product of degree -d on the homology of LM. We aim at identifying the three manifolds with “nontrivial” loop product. This is an application of some existing powerful tools in three-dimensional topology such as the prime decomposition, torus decomposition, Seifert fiber space theorem, torus theorem.     

On smooth Chas-Sullivan loop product in Quillen's geometric complex cobordism of Hilbert manifolds

In [A.J. Baker, C. Ozel, Complex cobordism of Hilbert manifolds with some applications to flag varieties, Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In [C. Ozel, On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups, in preparation], by using Quinn's Transversality Theorem [F. Quinn, Transversal approximation on Banach manifolds, Proc. Sympos. Pure Math. 15 (1970) 213-222], it has been shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It has been shown that the Thom isomorphism in this theory was satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps has been proved. In [M. Chas, D. Sullivan, String topology, math.GT/9911159, 1999], Chas and Sullivan described an intersection product on the homology of loop space LM. In [R.L. Cohen, J.D.S. Jones, A homotopy theoretic realization of string topology, math.GT/0107187, 2001], R. Cohen and J. Jones described a realization of the Chas-Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. In this paper, we will extend this product on cobordism and bordism theories.     

The loop product for 3-manifolds

Let M be a connected, closed, oriented and smooth manifold of dimension d. Let LM be the space of loops in M. Chas and Sullivan introduced the loop product, an associative product of degree ?d on the homology of LM. In this Note we aim at identifying 3-manifolds with “non-trivial” loop products. To cite this article: H. Abbaspour, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

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

Let Mⁿ be a closed, oriented, n-manifold, and LM its free loopspace. In [Chas and Sullivan, ‘String topology’,Ann. of Math., to appear] a commutative algebra structure inhomology, H_*(LM), and a Lie algebra structure in equivarianthomology , were defined.In this paper, we prove that these structures are homotopy invariantsin the following sense. Let f:M₁ M₂ be a homotopy equivalenceof closed, oriented n-manifolds. Then the induced equivalence,Lf:LM₁ LM₂ induces a ring isomorphism in homology, and an isomorphismof Lie algebras in equivariant homology. The analogous statementalso holds true for any generalized homology theory h_* thatsupports an orientation of the M_i. Received February 5, 2007.     

Homology of the double and the triple loop spaces¶of E6, E7, and E8

We study the mod p homology of the double and the triple loop spaces of exceptional Lie groups E ₆, E ₇, and E ₈ through the Eilenberg–Moore spectral sequence and the Serre spectral sequence using homology operations. The Bockstein actions on them are also determined. As a result, the Eilenberg–Moore spectral sequences of the path loop fibrations converging to H _*(Ω² G;? _p ) and H _*(Ω³ G;? _p ) collapse at the E ²-term for any compact simple Lie group G. Received: 11 November 1999     

Preduals of semigroup algebras

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C ₀(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak^*-continuous. Given a discrete semigroup S, the convolution algebra ℓ ¹(S) also carries a coproduct. In this paper we examine preduals for ℓ ¹(S) making both the product and the coproduct weak^*-continuous. Under certain conditions on S, we show that ℓ ¹(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ ¹(S) when S is either ℤ₊×ℤ or (ℕ,⋅).     

Algebraic geometry of topological spaces I

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case . We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C ^*-algebras, and for a homology theory of commutative algebras to vanish on C ^*-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil K-theory implies that commutative C ^*-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.     

Quantum cohomology of <Emphasis Type="Italic">G/P</Emphasis> and homology of affine Grassmannian

Let G be a simple and simply-connected complex algebraic group, P ⊂ G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH ^*(G/P) of a flag variety is, up to localization, a quotient of the homology H _*(Gr _G ) of the affine Grassmannian Gr _G of G. As a consequence, all three-point genus-zero Gromov–Witten invariants of G/P are identified with homology Schubert structure constants of H _*(Gr _G ), establishing the equivalence of the quantum and homology affine Schubert calculi.     

Maximal sets in α-recursion theory

Let α be an admissible ordinal, and leta ^* be the Σ₁-projectum ofa. Call an α-r.e. setM maximal if α→M is unbounded and for every α→r.e. setA, eitherA∩(α-M) or (α-A)∩(α-M) is bounded. Call and α-r.e. setM amaximal subset of α^* if α^*−M is undounded and for any α-r.e. setA, eitherA∩(α^*-M) or (⇌^*-A)∩(α^*-M) is unbounded in α^*. Sufficient conditions are given both for the existence of maximal sets, and for the existence of maximal subset of α^*. Necessary conditions for the existence of maximal sets are also given. In particular, if α ≧ ℵ ^L then it is shown that maximal sets do not exist. Research partially supported by NSF Grant GP-34088 X. Some of the results in this paper have been taken from the second author’s Ph. D. Thesis, written under the supervision of Gerald Sacks.     

The functor <Emphasis Type="Italic">A</Emphasis><Superscript>min</Superscript> for (<Emphasis Type="Italic">p</Emphasis> − 1)-cell complexes and EHP sequences

Let X be a co-H-space of (p − 1)-cell complex with all cells in even dimensions. Then the loop space ΩX admits a retract Ā ^min(X) that is the evaluation of the functor Ā ^min on X. In this paper, we determine the homology H _*(Ā ^min(X)) and give the EHP sequence for the spaces Ā ^min(X).     

Strongly and weakly almost periodic linear maps on semigroup algebras

Let S be a foundation locally compact topological semigroup. Two new topologies τ _c and τ _w are introduced on M _a (S)^*. We introduce τ _c and τ _w almost periodic functionals in M _a (S)^*. We study these classes and compare them with each other and with the norm almost periodic and weakly almost periodic functionals. For f∈M _a (S)^*, it is proved that T _f ∈ℬ(M _a (S),M _a (S)^*) is strong almost periodic if and only if f is τ _c -almost periodic. Indeed, we have obtained a generalization of a well known result of Crombez for locally compact group to a more general setting of foundation topological semigroups. Finally if P(S) (the set of all probability measures in M _a (S)) has the semiright invariant isometry property, it is shown that the set of τ _w -almost periodic functionals has a topological left invariant mean.     

Groupoid structure of Toeplitz algebra over L-shaped domain

The structure of the groupoidG associated with the ToeplitzC ^*-algebraC ^*(Ω) of the L-shaped domain is discussed. The detailed characterization ofM _∞ by the classification of the closed subgroup of the Euclidean space is presented. Project supported partially by the National Natural Science Foundation of China, Fok Yingtung Educational Foundation and the Foundation of the State Education Commission of China.     

Generalized Symplectic Action and Symplectomorphism Groups of Coadjoint Orbits

Let M be a quantizable symplectic manifold. If ψ_t is a loop in the group {Ham}(M) of Hamiltonian symplectomorphisms of M and A is a 2k-cycle in M, we define a symplectic action κ_A(ψ)∊ U(1) around ψ_t(A), which is invariant under deformations of ψ, and such that κ_A(ψ) depends only on the homology class of A. Using properties of κ_A( ) we determine a lower bound for ♯π₁(Ham(O)), where O is a quantizable coadjoint orbit of a compact Lie group. In particular we prove that ♯π₁(Ham(CPⁿ)) ≥ n+1. Mathematics Subject Classifications (2000): 53D05, 57S05, 57R17, 57T20.     

Pontrjagin–Thom maps and the homology of the moduli stack of stable curves

We study the singular homology (with field coefficients) of the moduli stack [`(\mathfrakM)]<sub>g, n</sub>{\overline{\mathfrak{M}}_{g, n}} of stable n-pointed complex curves of genus g. Each irreducible boundary component of [`(\mathfrakM)]_{g, n}{\overline{\mathfrak{M}}_{g, n}} determines via the Pontrjagin–Thom construction a map from [`(\mathfrakM)]<sub>g, n</sub>{\overline{\mathfrak{M}}_{g, n}} to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This detects many new torsion classes in the homology of [`(\mathfrakM)]_{g, n}{\overline{\mathfrak{M}}_{g, n}}.     

Homological stability for the mapping class groups of non-orientable surfaces

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of non-orientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum built from the canonical bundle over the Grassmannians of 2-planes in ℝ ⁿ⁺². In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i – this is the non-oriented analogue of the Mumford conjecture.     

Chern-Simons invariant and conformal embedding of a 3-manifold

This note studies the Chern-Simons invariant of a closed oriented Riemannian 3-manifold M. The first achievement is to establish the formula CS（e） - CS（e） = degA, where e and e are two （global） frames of M, and A ： M → SO（3） is the ＂difference＂ map. An interesting phenomenon is that the ＂jumps＂ of the Chern-Simons integrals for various frames of many 3-manifolds are at least two, instead of one. The second purpose is to give an explicit representation of CS（e＋） and CS（e_）, where e＋ and e_ are the ＂left＂ and ＂right＂ quaternionic frames on M3 induced from an immersion M^3 → E^4, respectively. Consequently we find many metrics on S^3 （Berger spheres） so that they can not be conformally embedded in E^4.     

Codepth Two and Related Topics

A depth two extension A | B is shown to be weak depth two over its double centralizer V _A (V _A (B)) if this is separable over B. We consider various examples and non-examples of depth one and two properties. Depth two and its relationship to direct and tensor product of algebras as well as cup product of relative Hochschild cochains is examined. Section 6 introduces a notion of codepth two coalgebra homomorphism g : C → D, dual to a depth two algebra homomorphism. It is shown that the endomorphism ring of bicomodule endomorphisms End ^D C ^D forms a right bialgebroid over the centralizer subalgebra g ^* : D ^* → C ^* of the dual algebra C ^*. Dedicated to Daniel Kastler on his eightieth birthday.     

On the filter of computably enumerable supersets of an r-maximal set

We study the filter ℒ^*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ^*(A) is still Σ⁰ ₃-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ^*(A). Received: 6 November 1999 / Revised version: 10 March 2000 /?Published online: 18 May 2001