Torsion-free Part Research Articles

Abelianization of the unit group of an integral group ring

For a finite group $G$ and $U: = U(\mathbb{Z}G)$, the group of units of the integral group ring of $G$, we study the implications of the structure of $G$ on the abelianization $U/U'$ of $U$. We pose questions on the connections between the exponent of $G/G'$ and the exponent of $U/U'$ as well as between the ranks of the torsion-free parts of $Z(U)$, the center of $U$, and $U/U'$. We show that the units originating from known generic constructions of units in $\mathbb{Z}G$ are well-behaved under the projection from $U$ to $U/U'$ and that our questions have a positive answer for many examples. We then exhibit an explicit example which shows that the general statement on the torsion-free part does not hold, which also answers questions from [BJJ$^+$18].

On the conical Novikov homology

Let $\omega$ be a Morse form on a manifold $M$. Let $p:\hat M\to M$ be a regular covering with structure group $G$, such that $p^*([\omega])=0$. Let $\xi:G\to\mathbf{R}$ be the corresponding period homomorphism. Denote by ${\hat \Lambda}_\xi$ the Novikov completion of the group ring $\mathbf{Z} G$. Choose a transverse $\omega$-gradient $v$. Counting the flow lines of $v$ one defines the Novikov complex $\mathcal{N}_*$ freely generated over ${\hat \Lambda}_\xi$ by the set of zeroes of $\omega$. In this paper we introduce a refinement of this construction. We define a subring $\hat\Lambda_\Gamma$ of ${\hat \Lambda}_\xi$ and show that the Novikov complex $\mathcal{N}_*$ is defined actually over $\hat\Lambda_\Gamma$ and computes the homology of the chain complex $C_*(\hat M)\underset{\Lambda}{\otimes}\hat\Lambda_\Gamma $. When $G\approx\mathbf{Z}^2$, and the irrationality degree of $\xi$ equals 2, the ring $\hat\Lambda_\Gamma$ is isomorphic to the ring of series in $2$ variables $x, y$ of the form $\sum_{r\in\mathbf{N}} a_r x^{n_r}y^{m_r}$ where $a_r, n_r, m_r\in\mathbf{Z}$ and both $n_r, \ m_r$ converge to $\infty$ when $r\to \infty$. The algebraic part of the proof is based on a suitable generalization of the classical algorithm of approximating irrational numbers by rationals. The geometric part is a straightforward generalization of the author's proof of the particular case of this theorem concerning the circle-valued Morse maps. In Appendix 1 we give an overview of E. Pitcher's work on circle-valued Morse theory (1939). We show that Pitcher's lower bounds for the number of critical points of a circle-valued Morse map coincide with the torsion-free part of the Novikov inequalities. In Appendix 2 we construct a circle-valued Morse map and its gradient such that its unique Novikov incidence coefficient is a power series in one variable with an arbitrarily small convergence radius.

On BP⟨2⟩–cooperations

In this paper we develop techniques to compute the cooperations algebra for the second truncated Brown-Peterson spectrum $\tBP{2}$. We prove that the cooperations algebra $\tBP{2}_*\tBP{2}$ decomposes as a direct some of a $\F_2$-vector space concentrated in Adams filtration 0 and a $\F_2[v_0,v_1,v_2]$-module which is concentrated in even degrees and $v_2$-torsion free. A recursive procedure is also developed to provide an basis of the $v_2$-torsion free part.

Homomorphically Stable Abelian Groups

A group is said to be homomorphically stable with respect to another group if the union of the homomorphic images of the first group in the second group is a subgroup of the second group. A group is said to be homomorphically stable if it is homomorphically stable with respect to every group. It is shown that a group is homomorphically stable if it is homomorphically stable with respect to its double direct sum. In particular, given any group, the direct sum and the direct product of infinitely many copies of this group are homomorphically stable; all endocyclic groups are homomorphically stable as well. Necessary and sufficient conditions for the homomorphic stability of a fully transitive torsion-free group are found. It is proved that a group is homomorphically stable if and only if so is its reduced part, and a split group is homomorphically stable if and only if so is its torsion-free part. It is shown that every group is homomorphically stable with respect to every periodic group.

Вычисление рангов групп центральных единиц целочисленных групповых колец конечных групп

The study of central units (central invertible elements) of integral group rings is encountered to difficult calculations almost everywhere, both in the case of finding of individual central unit and in the case of describing of group of central elements. By virtue of torsion part of central unit group is trivial (up to sign those are elements of center group) it is more interesting to find data about torsion free part that is direct product infinite cyclic groups. The number of such infinite factors is the rank of central unit group. Therefore the ranks of central unit groups of integral group rings of finite groups are the very important characteristic those groups. So that the computation ranks of central unit groups has big interest for study of central unit groups. In the paper we point out the formulas for computation of ranks in general case and some important particular cases. On the base of those formulas we compute the ranks in quite large ranges. We used computer algebra system GAP. The results are shown on tables and graph.

Construction of units in ZC_{24}

Torsion-free part of the unit group of ZC24 is generated by 5 units. In this paper, we firstly will give a parametrization of torsion-free units in ZC24 and later we shall characterize the unit group of ZC24 by using some ring extensions. Mathematics Subject Classification: 16S34, 16U60

Formula omitted]-theory of locally finite graph [formula omitted]-algebras

We calculate the K-theory of the Cuntz–Krieger algebra OE associated with an infinite, locally finite graph, via the Bass–Hashimoto operator. The formulae we get express the Grothendieck group and the Whitehead group in purely graph theoretic terms.We consider the category of finite (black-and-white, bi-directed) subgraphs with certain graph homomorphisms and construct a continuous functor to abelian groups. In this category K0 is an inductive limit of K-groups of finite graphs, which were calculated in Cornelissen et al. (2008) [3].In the case of an infinite graph with the finite Betti number we obtain the formula for the Grothendieck group K0(OE)=Zβ(E)+γ(E), where β(E) is the first Betti number and γ(E) is the valency number of the graph E. We note that in the infinite case the torsion part of K0, which is present in the case of a finite graph, vanishes. The Whitehead group depends only on the first Betti number: K1(OE)=Zβ(E). These allow us to provide a counterexample to the fact, which holds for finite graphs, that K1(OE) is the torsion free part of K0(OE).

Torsion-free endotrivial modules

Let G be a finite group and let T(G) be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We investigate the torsion-free part TF(G) of the group T(G) and look for generators of TF(G). We describe three methods for obtaining generators. Each of them only gives partial answers to the question but we obtain more precise results in some specific cases. We also conjecture that TF(G) can be generated by modules belonging to the principal block and we prove the conjecture in some cases.

K-theory for ring C*-algebras attached to function fields with only one infinite place

AbstractWe study the K-theory of ring C*-algebras associated to rings of integers in global function fields with only a single infinite place. First, we compute the torsion-free part of the K-groups of these ring C*-algebras. Secondly, we show that, under a certain primeness condition, the torsion part of K-theory determines the inertia degrees at infinity of our function fields.

Units in real cyclic fields

Let $N/\mathbb{Q}$ be a real cyclic and tame extension of prime degree $l$ with $\Gamma=\mathscr{G}al(N/\mathbb{Q})$. We give the Hom description of the class of the torsion-free part of the group of units in $N$ in the class group of the order $\mathbb{Z}\Gamma/ (\sum_{\gamma \in \Gamma}\gamma )$. This representation depends only on the structure of the ideal class group of $N$ and determines the Galois module structure of the torsion-free part of the group of units in $N$ as an ideal of the $l$th cyclotomic field. Using this approach we derive necessary and sufficient conditions for all real and tame cyclic fields of prime degree to have Minkowski units. We extend also the class of known cyclic real fields with Minkowski units.

The torsion-free part of the Ziegler spectrum of the Klein four group

We will describe the torsion-free part of the Ziegler spectrum, both the points and the topology, over the integral group ring of the Klein group. For instance we will show that the Cantor–Bendixson rank of this space is equal to 3.

The group of endotrivial modules for the symmetric and alternating groups

AbstractWe complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n ≥ p2, the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelian of rank 1 if n ≥ p2 + p and has rank 2 if p2 ≤ n < p2 + p. This completes the work begun earlier by Carlson, Mazza and Nakano.

The chromatic polynomial of fatgraphs and its categorification

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homology to show that the torsion-free part of our chromatic homology is independent of the choice of planar embedding of a graph. We extend our construction and categorify the Bollobás–Riordan polynomial (a generalization of the Tutte polynomial to embedded graphs). We prove that both our chromatic homology and the Khovanov homology of an associated link can be recovered from this categorification.

Torsion Theories for Algebras of Affiliated Operators of Finite Von Neumann Algebras

The dimension of any module over an algebra of affiliated operators ${\mathcal U}$ of a finite von Neumann algebra ${\mathcal A}$ is defined using a trace on ${\mathcal A}.$ All zero-dimensional ${\mathcal U}$-modules constitute the torsion class of torsion theory $(\mathrm{{\bf T}},\mathrm{{\bf P}})$. We show that every finitely generated ${\mathcal U}$-module splits as the direct sum of torsion and torsion-free part. Moreover, we prove that the theory $(\mathrm{{\bf T}},\mathrm{{\bf P}})$ coincides with the theory of bounded and unbounded modules and also with the Lambek and Goldie torsion theories. Lastly, we use the introduced torsion theories to give the necessary and sufficient conditions for ${\mathcal U}$ to be semisimple.

The Jacobian of a nonorientable Klein surface

Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surfaceY is constructed. The Jacobian is identified with the dual of the space of all harmonic real one-forms onY quotiented by the torsion-free part of the first integral homology ofY. Denote byX the double cover ofY given by orientation. The Jacobian ofY is identified with the space of all degree zero holomorphic line bundlesL overX with the property thatL is isomorphic to σ*/-L, where σ is the involution ofX.

On Gusarov's groups of knots

We give a construction of Gusarov's groups [Gscr ] n of knots based on pure braid commutators, and show that any element of [Gscr ] n is represented by an infinite number of prime alternating knots of braid index less than or equal to n +1. We also study [Vscr ] n , the torsion-free part of [Gscr ] n , which is the group of equivalence classes of knots which cannot be distinguished by any rational Vassiliev invariant of order less than or equal to n . Generalizing the Gusarov–Ohyama definition of n -triviality, we give a characterization of the elements of the n th group of the lower central series of an arbitrary group.

Topological invariants of a fan associated to a toric variety

Associated to a toric variety X of dimension r over a field k is a fan Δ on R1. The fan Δ is a finite set of cones which are in one-to-one correspondence with the orbits of the torus action on X. The fan Δ inherits the Zariski topology from X. In this article some cohomological invariants of X are studied in terms of whether or not they depend only on Δ and not k. Secondly some numerical invariants of X are studied in terms of whether or not they are topological invariants of the fan Δ. That is, whether or not they depend only on the finite topological space defined on Δ. The invariants with which we are mostly concerned are the class group of Weil divisors, the Picard group, the Brauer group and the dimensions of the torsion free part of the etale cohomology groups with coefficients in the sheaf of units. The notion of an open neighborhood of a fan is introduced and examples are given for which the above invariants are sufficiently fine to give nontrivial stratifications of an open neighborhood of a fan all of whose maximal cones are nonsimplicia.

Periodic maps on simply connected four-manifolds

IN THIS paper we describe certain invariants of periodic maps on closed, simply connected four-manifolds with the purpose of classifying such maps up to homeomorphism. The manifolds themselves are classified by Freedman [9] in terms of the intersection pairing on second homology and the Kirby--Sicbcnmann invariant. Gcncralizing Freedman’s result to some nonsimply connected four-manifolds Hambleton and Kreck [IO] have recently shown that in the case of a cyclic fundamental group of odd order the manifold is determined up to homeomorphism by the same two invariants togcthcr with the fundamental group (with the intcrscction pairing being understood now as dcfincd on the torsion free part of the second homology group). A similar statement for cvcn order groups fails to bc true. although its failure is less dramatic for algebraic surfaces as follows from the work by the same authors [I I]. From the point of view of the fundamental group acting on the universal cover these results concern the topological classification of fret periodic maps on simply conncctcd four-manifolds. In gcncral, howcvcr. a periodic map is allowed to have fixed points or periodic points of period smaller than the map period. When this happens, the quotient orbifold has singularities. We shall show that in the cast of isolated singularities one can still classify periodic maps by means of some invariants associated with this orbifold, although in the singular case one must also take into account the torsion part of its second homology. In the more rigid situation of a free periodic map no torsion information is needed. This can bc partially accounted for by the fact that many four-manifolds do not support any such maps. On the other hand, by a result of Edmonds [5]. every closed, simply connected fourmanifold supports infinitely many periodic maps with isolated fixed points. The paper is organized in four sections. In Section 2 we discuss the homotopy classification of locally linear, pseudofree actions of finite groups on simply connected fourmanifolds. There we introduce some terminology and prepare the obstruction theory needed for the results of the next two sections. Our main result is the topological classification of semifree periodic maps of odd period. This is the content of Theorem 3.1. A related but somewhat weaker result for periodic maps of even period is contained in Theorem 3.2. As an application of our techniques, we prove in Section 4 that any locally linear, pscudofree action of a finite group on the complex projective plane is topologically conjugate to a linear action. This last statcmcnt about actions on CP’ has also been asserted in [I93 in the context of finite group actions on the Chern manifold. There we concentrated our efforts on actions without fixed points referring the reader for the proof in the semifrce case to the combined results of [7] and [ 121. This point requires now further elaboration for maps of composite period as the authors of [ 121 have meanwhile added to their result a technical hypothesis of

On the Kummer congruences and the stable homotopy of 𝐵U

We study the torsion-free part of the stable homotopy groups of the space B U BU , by considering upper and lower bounds. The upper bound is furnished by the ring P K ∗ ( B U ) P{K_{\ast }}(BU) of coaction primitives into which π ∗ S ( B U ) \pi _{\ast }^S(BU) is mapped by the complex K K -theoretic Hurewicz homomorphism \[ π ∗ S ( B U ) → P K ∗ ( B U ) . \pi _{\ast }^S(BU) \to P{K_{\ast }}(BU). \] We characterize P K ∗ ( B U ) P{K_{\ast }}(BU) in terms of symmetric numerical polynomials and describe systematic families of elements by utilizing the classical Kummer congruences among the Bernoulli numbers. For a lower bound we choose the ring of those framed bordism classes which may be represented by singular hypersurfaces in B U BU . From among these we define families of classes constructed from regular neighborhoods of embeddings of iterated Thom complexes in Euclidean space. Employing techniques of duality theory, we deduce that these two families correspond, except possibly in the lowest dimensions, under the Hurewicz homomorphism, which thus provides a link between the algebra and the geometry. In the course of this work we greatly extend certain e e -invariant calculations of J. F. Adams.

On the structure of Ext(A,Z) in ZFC+

A fundamental problem in the theory of abelian groups is to determine the structure of Ext(A, Z) for arbitrary abelian groups A. This problem was raised by L. Fuchs in 1958, and since then has been the center of considerable activity and progress.We briefly summarize the present state of this problem. It is a well-known fact thatwhere tA denotes the torsion subgroup of A. Thus the structure problem for Ext(A, Z) breakdown to the two distinct cases, torsion and torsion free groups. For a torsion group T,which is compact and reduced, and its structure is known explicitly [12].For torsion free A, Ext(A, Z) is divisible; hence it has a unique representationThus Ext(A, Z) is characterized by countably many cardinal numbers, which we denote as follows: ν0(A) is the rank of the torsion free part of Ext(A, Z), and νp(A) are the ranks of the p-primary parts of Ext(A, Z), Extp(A, Z).If A is free it is an elementary fact that Ext(A, Z) = 0. The second named author has shown [16] that in the presence of V = L the converse is also true. For countable torsion free, nonfree A, C. Jensen [13] has shown that νp(A) is either finite or and νp(A) ≤ ν0(A). Therefore, the case for uncountable, nonfree, torsion free groups A remains to be studied.