Degree One Maps Between Geometric 3-Manifolds

Let M n + 3 {M^{n + 3}} be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold K 3 {K^3} with an effective circle action of hyperbolic type. Assume that the fiber N n {N^n} in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set S TOP ( N n × D k , ∂ ) {S_{\text {TOP}}}\,({N^n} \times {D^k},\partial ) of topological surgery vanishes for sufficiently large k k . Then the projective class and Whitehead groups of M M vanish and S TOP ( M n + 3 × D k , ∂ ) = 0 {S_{\text {TOP}}}\,({M^{n + 3}}\, \times \, {D^k},\partial ) = 0 if n + k ⩾ 3 n + k \geqslant 3 or if K 3 {K^3} is closed and n = 2 n = 2 . The UNil \text {UNil} groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that UNil j {\text {UNil}}_j vanishes in spite of the failure of Cappell’s sufficient condition.

Let M M and N N be two compact orientable 3 3 -manifolds, we say that M ≥ N M \geq N , if there is a degree one map from M M to N N . This gives a way to measure the complexity of 3 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 3 -manifolds" must eventually become homotopy equivalences.

We introduce and study trace fields and invariant trace fields of SL2(C) and PSL2(C) representations of 3-manifold groups. We give conditions on such fields which imply that the underlying 3-manifold is virtually Haken or has virtually positive or 1 first Betti number. In particular, we define the notion of an algebraically trace-proper surface subgroup in a 3-manifold group, and show that any closed orientable irreducible 3-manifold with such a surface subgroup is virtually Haken. We give infinitely many families of closed orientable hyperbolic non-Haken 3-manifolds with algebraically trace-proper surface subgroups. In this paper a 3-manifold is always assumed to be connected and orientable, without loss of generality for the problems to be considered. A 3-manifold is said to be Haken if it is compact, irreducible, and contains a properly embedded incompressible surface. A 3-manifold is said to be virtually Haken if it has a finite cover which is Haken. The well-known virtually Haken conjecture states that every closed and irreducible 3-manifold with infinite fundamental group is virtually Haken. It is also conjectured that every closed Haken 3-manifold has virtually positive first Betti number, that is, the manifold has a finite cover which has positive first Betti number. If M is a closed hyperbolic Haken 3-manifold, then it is further conjectured that M has virtually 1 first Betti number, which means that for any positive integer n, there is a finite cover Mn of M such that the first Betti number of Mn is larger than n. These conjectures are fundamental and difficult issues in 3-manifold topology. In this paper we provide some information about these conjectures by studying traces of representations of 3-manifold groups into SL2ðCÞ or PSL2ðCÞ: For a group G and a representation r : G ! SL2ðCÞ; we define the trace field Kr of r to be the field generated by the traces of all the matrices r (g), g [ G; over the base field Q of rational numbers, that is, Kr ¼ Qðtrðr ðgÞÞ; g [ G Þ:

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.

In the context of Thurstons geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We show that non-geometric Haken manifolds generically, but not always, admit such metrics. More precisely, we prove that a Haken manifold with, possibly empty, boundary of zero Euler characteristic admits metrics of nonpositive curvature if the boundary is non-empty or if at least one atoroidal component occurs in its canonical topological decomposition. Our arguments are based on Thurstons Hyperbolisation Theorem. We give examples of closed graph-manifolds with linear gluing graph and arbitrarily many Seifert components which do not admit metrics of nonpositive curvature.

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

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.

We study the etale homotopy theory of Brauer-Severi varieties over fields of characteristic 0. We prove that the induced Galois representations on geometric homotopy invariants (e.g., l-adic cohomology or higher homotopy groups) are all isomorphic for Brauer-Severi varieties of the same dimension. If the base field has cohomological dimension smaller or equal 2 then we can show more in the case of Brauer-Severi curves: There is even an isomorphism between the Hochschild-Serre spectral sequences computing cohomology with local coefficients. Further, we study homotopy rational and homotopy fixed points on Brauer-Severi varieties and their connections to genuine rational points. In particular, we show that under a suitable assumption on the first profinite Chern class map an analogue of the weak section conjecture for Brauer-Severi varieties turns out to be true. We can give a counter example to this analogue without the extra assumption over p-adic local fields.

Polyhedra for which every homotopy domination over itself is a homotopy equivalence

We prove that a leafwise homotopy equivalence between compact foliated manifolds induces a well defined bounded operator between all Sobolov spaces of leafwise (for the natural foliations of the graphs of the original foliations) differential forms with coefficients in a leafwise flat bundle. We further prove that the associated map on the leafwise reduced L2 cohomology is an isomorphism which only depends on the leafwise homotopy class of the homotopy equivalence.

The homotopy classes of Lorentz signature metric tensor fields on parallelisable four-manifolds are classified for both compact and non-compact manifolds. It is demonstrated that the homotopy classes form Abelian groups whose generators correspond to generalised elementary 'kinks', and their extension structures are analysed. It is also demonstrated that all odd kink states have the fermionic property of being odd under 360 degrees rotations. Speculations on the physical interpretation of four-dimensional kink states are made which generalise those of Shastri, Williams and Zvengrowski (1980) from three to four dimensions.

Driven by environmental concerns and fluctuations of electric energy cost, the integration of distributed generators (DGs) into distribution networks close to the loads, has gained its importance and emerged as complementary to the conventional central power plants. However, the proliferation of distributed generators (DGs) has altered a distribution network from a passive system to an active one. Hence, modifications in the existing distribution power flow methods are needed to account for a DG unit, which can operate either as a PV or PQ bus. Among these methods, the Sequential Power Flow (SPF) method can readily accommodate for PV buses as long as the DG models in the sequence-component frame can be formulated. However, the convergence of SPF is heavily dependent on the choice of initial conditions. Moreover, when the resistance-to-reactance ratio (or R/X) is high, SPF will also diverge. This paper presents a new robust three-phase distribution power flow, which incorporates a homotopic method into the SPF method to overcome these problems. An IEEE benchmark system is used to test the validity and robustness of the proposed algorithm