Structure sets vanish for certain bundles over Seifert manifolds

We give a characterization of k-dimensional (k > 1) Menger manifolds admitting boundaries in the sense of Chapman and Siebenmann.In [5] Chapman and Siebenmann considered the problem of putting a boundary on a Hilbert cube manifold (similar problems in the cases of smooth and piecewise linear manifolds were considered in [2,14]; a parametrical version of the above problem was considered in [13]).It was proved there that if a Q-manifold M satisfies certain minimal necessary homotopy-theoretic conditions (finite type and tameness at oo ), then there are two obstructions o0O(M) and Too(M) to M having a boundary.The first one is an element of the group lim{J?67ri(M -A) : A c M compact}, where %nx is the projective class group functor.If Ooo(M) = 0, then the second obstruction can be defined as an element of the first derived limit of the inverse system lim {W hnx (M -A) : A M compact} , where Whnx is the Whitehead group functor.Further, it was proved in [5] that the different boundaries that can be put on M constitute a whole shape class and that a classification of all possible ways of putting boundaries on M can be done in terms of the group lim {Whnx (M -A) : Ac.M compact}.In the present paper we carry out a similar program for the problem of putting boundaries on pn+x -manifolds, where pn+x denotes the (n + 1 )-dimensional universal Menger compactum (a p"+x -manifold M admits a boundary if there exists a compact p"+x-manifold N such that M = N-Z , where Z is a Z-set in N ; in this case we shall say that N is a compactification of M corresponding to the boundary Z , and conversely, Z is a boundary of M corresponding to the compactification N. We recall also that a closed subset Z of a space X is said to be a Z-set if for each open cover ^ e cov(X) there is a map f:X-*X-Z ^-close to the identity map of X ).Having in mind a deep analogy between the theories of pn+x -manifolds and Q-manifolds [1,[6][7][8][9] it is not surprising that the corresponding results are valid in the case of pn+xmanifolds as well.However, it should be observed that the situation in the last case is much simpler.For example, the analogies of the above-described

The Projective Class Group and the Whitehead Group

Let M and N be two compact orientable 3-manifolds, we say that M ≥ N, if there is a degree one map from M to N. This gives a way to measure the complexity of 3-manifolds. The main purpose of this paper is to give a positive answer to the following conjecture: if there is an infinite sequence of degree one maps between Haken manifolds, then eventually all the manifolds are homeomorphic to each other. More generally, we prove a theorem which says that any infinite sequence of degree one maps between the so-called «geometric 3-manifolds» must eventually become homotopy equivalences

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$-forms. We show that the spectral function of $\Box^{(q)}_b$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$, then the spectrum of $\Box^{(q)}_b$ in $]0,\infty[$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation of the associated Szeg\"o kernel admits an asymptotic expansion under a local closed range condition. As applications, we establish the Szeg\"o kernel asymptotic expansions on some weakly pseudoconvex CR manifolds and on CR manifolds with transversal CR $S^1$ actions. By using these asymptotics, we establish some local embedding theorems on CR manifolds and we give an analytic proof of a theorem of Lempert asserting that a compact strictly pseudoconvex CR manifold of dimension three with a transversal CR $S^1$ action can be CR embedded into $\mathbb{C}^N$, for some $N\in\mathbb N$.

We consider a compact orientable Einstein manifold of even dimension n=2m, which is k-pinched. If k>[m(m−1) 2 (2m−1) 2 (2m−1)+3], then there exists no element, different from zero of the H 2 (M,R) such that its exterior m-power belongs to a zero class. This result has some applications to the topological product of manifold.

Projective modules and class groups

This note gives a complete description of the cohomology algebra of any orientable Seifert manifold with ℤ/p coefficients, for an arbitrary prime p. As an application, the existence of a degree one map from an orientable Seifert manifold onto a lens space

It has been conjectured that, on a compact 3-dimensional orientable manifold, a critical point of the total scalar curvature restricted to the space of constant scalar curvature metrics of unit volume is Einstein. In this paper we prove this conjecture under a condition that ker , which generalizes the previous partial results.

Suppose $f\colon M\to M$ is a $C^{1+\alpha}$ $(\alpha>0)$ diffeomorphism on a compact smooth orientable manifold $M$ of dimension 2, and let $\mu_\Psi$ be an equilibrium measure for a Hölder-continuous potential $\Psi\colon M\to \mathbb R$. We show that if $\mu_\Psi$ has positive measure-theoretic entropy, then $f$ is measure-theoretically isomorphic mod $\mu_\Psi$ to the product of a Bernoulli scheme and a finite rotation.

Let M be a compact connected Riemannian manifold whose sectional curvature values are all nonpositive. Let T denote the fundamental group of M. We prove that any homotopy equivalence : N -> M from a compact closed manifold TV is homotopic to a homeomorphism, provided that m > 5 where m = dim M . We show that the surgery L-group L k+m (T, w { ) is isomorphic to the set of homotopy classes of k k

We reconsider Chern-Simons gauge theory on a Seifert manifold M (the total space of a nontrivial circle bundle over a Riemann surface Σ). When M is a Seifert manifold, Lawrence and Rozansky have shown from the exact solution of Chern-Simons theory that the partition function has a remarkably simple structure and can be rewritten entirely as a sum of local contributions from the flat connections on M. We explain how this empirical fact follows from the technique of non-abelian localization as applied to the Chern-Simons path integral. In the process, we show that the partition function of Chern-Simons theory on M admits a topological interpretation in terms of the equivariant cohomology of the moduli space of flat connections on M.

Chapter 26 - Wall's Finiteness Obstruction

In this note we give a necessary and sufficient condition for the existence of a fiber preserving branched covering between two closed, orientable Seifert manifolds (for sufficiency we need the additional assumption that the genus of the base orbifold of the target manifold > 1). Combining this with two theorems of Rong we get a necessary and sufficient condition for the existence of a nonzero degree map between two such manifolds.

A necessary and sufficient condition for a Heegaard splitting to be keen weakly reducible

Let A A be a Noetherian Cohen-Macaulay domain, b b , c 1 c_1 , … \dots , c g c_g an A A -sequence, J J = ( b , c 1 , … , c g ) A (b,c_1,\dots ,c_g)A , and B B = A [ J / b ] A[J/b] . Then B B is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets Ass B ⁡ ( B / b B ) \operatorname {Ass}_B(B/bB) and Ass A ⁡ ( A / J ) \operatorname {Ass}_A(A/J) , and each q q ∈ \in Ass A ⁡ ( A / J ) \operatorname {Ass}_A(A/J) has height g + 1 g+1 . If B B does not have unique factorization, then some height-one prime ideals P P of B B are not principal. These primes are identified in terms of J J and P ∩ A P \cap A , and we consider the question of how far from principal they can be. If A A is integrally closed, necessary and sufficient conditions are given for B B to be integrally closed, and sufficient conditions are given for B B to be a UFD or a Krull domain whose class group is torsion, finite, or finite cyclic. It is shown that if P P is a height-one prime ideal of B B , then P ∩ A P \cap A also has height one if and only if b b ∉ \notin P P and thus P ∩ A P \cap A has height one for all but finitely many of the height-one primes P P of B B . If A A has unique factorization, a description is given of whether or not such a prime P P is a principal prime ideal, or has a principal primary ideal, in terms of properties of P ∩ A P \cap A . A similar description is also given for the height-one prime ideals P P of B B with P ∩ A P \cap A of height greater than one, if the prime factors of b b satisfy a mild condition. If A A is a UFD and b b is a power of a prime element, then B B is a Krull domain with torsion class group if and only if J J is primary and integrally closed, and if this holds, then B B has finite cyclic class group. Also, if J J is not primary, then for each height-one prime ideal p p contained in at least one, but not all, prime divisors of J J , it holds that the height-one prime p A [ 1 / b ] ∩ B pA[1/b] \cap B has no principal primary ideals. This applies in particular to the Rees ring R {\mathbf R} = = A [ 1 / t , t J ] A[1/t, tJ] . As an application of these results, it is shown how to construct for any finitely generated abelian group G G , a monoidal transform B B = A [ J / b ] A[J/b] such that A A is a UFD, B B is Cohen-Macaulay and integrally closed, and G G ≅ \cong Cl ⁡ ( B ) \operatorname {Cl}(B) , the divisor class group of B B .