natural homomorphism
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2021 ◽  
pp. 2150061
Author(s):  
Xiongwei Cai

Given a crossed module of groupoids [Formula: see text], we construct (1) a natural homomorphism from the product groupoid [Formula: see text] to the crossed product groupoid [Formula: see text] and (2) a transgression map from the singular cohomology [Formula: see text] of the nerve of the groupoid [Formula: see text] to the singular cohomology [Formula: see text] of the nerve of the crossed product groupoid [Formula: see text]. The latter turns out to be identical to the transgression map obtained by Tu–Xu in their study of equivariant [Formula: see text]-theory.


Author(s):  
Fabian Haiden

AbstractWe compare two associative algebras which encode the “quantum topology” of Legendrian curves in contact threefolds of product type $$S\times {\mathbb {R}}$$ S × R . The first is the skein algebra of graded Legendrian links and the second is the Hall algebra of the Fukaya category of S. We construct a natural homomorphism from the former to the latter, which we show is an isomorphism if S is a disk with marked points and injective if S is the annulus.


2019 ◽  
Vol 29 (08) ◽  
pp. 1451-1466
Author(s):  
D. Gonçalves ◽  
T. Nasybullov

For [Formula: see text] denote by [Formula: see text] the free group on [Formula: see text] generators and let [Formula: see text]. For [Formula: see text] and elements [Formula: see text], we study orientable quadratic equations of the form [Formula: see text] with unknowns [Formula: see text] and provide explicit solutions for them for the minimal possible number [Formula: see text]. In the particular case when [Formula: see text], [Formula: see text] for [Formula: see text] and [Formula: see text] the minimal number which satisfies [Formula: see text], we provide two types of solutions depending on the image of the subgroup [Formula: see text] generated by the solution under the natural homomorphism [Formula: see text]: the first solution, which is called a primitive solution, satisfies [Formula: see text], the second solution satisfies [Formula: see text]. We also provide an explicit solution of the equation [Formula: see text] for [Formula: see text] in [Formula: see text], and prove that if [Formula: see text], then every solution of this equation is primitive. As a geometrical consequence, for every solution, we obtain a map [Formula: see text] from the orientable surface [Formula: see text] of genus [Formula: see text] to the torus [Formula: see text] which has the minimal number of roots among all maps from the homotopy class of [Formula: see text]. Depending on the number [Formula: see text], such maps have fundamentally different geometric properties: in some cases, they satisfy the Wecken property and in other cases not.


2018 ◽  
Vol 21 (3) ◽  
pp. 511-530
Author(s):  
Jonathan A. Hillman

Abstract We show that if π is the fundamental group of a 4-dimensional infrasolvmanifold then {-2\leq\mathrm{def}(\pi)\leq 0} , and give examples realizing each value allowed by our constraints, for each possible value of the rank of {\pi/\pi^{\prime}} . We also consider the abstract commensurators of such groups. Finally, we show that if G is a finitely generated group, the kernel of the natural homomorphism from G to its abstract commensurator {\mathrm{Comm}(G)} is locally nilpotent by locally finite, and is finite if {\mathrm{def}(G)>1} .


2015 ◽  
Vol 14 (08) ◽  
pp. 1550122
Author(s):  
Atiyeh Pour Eshmanan Talemi ◽  
Abolfazl Tehranian

Let (R, 𝔪, k) be a complete Gorenstein local ring of dimension n. Let [Formula: see text] be the local cohomology module with respect to a pair of ideals I, J and [Formula: see text]. In this paper we will show that the endomorphism ring [Formula: see text] is a commutative ring. In particular if [Formula: see text] for all i ≠ t, then B is isomorphic to R. Also we prove that, B is a finite R-module if and only if [Formula: see text] is an Artinian R-module, where d = n - t. Moreover we will show that in the case that [Formula: see text] for all i ≠ t the natural homomorphism [Formula: see text] is nonzero which gives a positive answer to a conjecture due to Hellus–Schenzel (see [On cohomologically complete intersections, J. Algebra 320 (2008) 3733–3748]).


2008 ◽  
Vol 15 (03) ◽  
pp. 457-462 ◽  
Author(s):  
A. Mafi ◽  
H. Saremi

Let R be a commutative Noetherian local ring, 𝔞 an ideal of R, and M a finitely generated generalized f-module. Let t be a positive integer such that [Formula: see text] and t > dim M - dim M/𝔞M. In this paper, we prove that there exists an ideal 𝔟 ⊇ 𝔞 such that (1) dim M - dim M/𝔟M = t; and (2) the natural homomorphism [Formula: see text] is an isomorphism for all i > t and it is surjective for i = t. Also, we show that if [Formula: see text] is a finite set for all i < t, then there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t.


2008 ◽  
Vol 18 (05) ◽  
pp. 803-823 ◽  
Author(s):  
HANS-JOACHIM BAUES ◽  
ROMAN MIKHAILOV

We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group π3. This generalizes a result of Gutierrez–Ratcliffe who relate the intersection of two subgroups with the computation of π2. Let K be a two-dimensional CW-complex with subcomplexes K1, K2, K3 such that K = K1 ∪ K2 ∪ K3 and K1 ∩ K2 ∩ K3 is the 1-skeleton K1 of K. We construct a natural homomorphism of π1(K)-modules [Formula: see text] where Ri = ker {π1(K1) → π1(Ki)}, i = 1,2,3 and the action of π1(K) = F/R1R2R3 on the right-hand abelian group is defined via conjugation in F. In certain cases, the defined map is an isomorphism. Finally, we discuss certain applications of the above map to group homology.


2007 ◽  
Vol 72 (4) ◽  
pp. 1177-1193 ◽  
Author(s):  
Alessandro Berarducci

AbstractBy recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal “infinitesimal subgroup” G00 such that the quotient G/G00, equipped with the “logic topology”, is a compact (real) Lie group. Our first result is that the functor G ↦ G/G00 sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G00 and the o-minimal spectrum of G. We prove that G/G00 is a topological quotient of . We thus obtain a natural homomorphism Ψ* from the cohomology of G/G00 to the (Čech-)cohomology of . We show that if G00 satisfies a suitable contractibility conjecture then is acyclic in Čech cohomology and Ψ is an isomorphism. Finally we prove the conjecture in some special cases.


2006 ◽  
Vol 3 (3) ◽  
pp. 465-469
Author(s):  
Baghdad Science Journal

Let/. It :0 ---0 G be any two self maps of a compact connected oriented Lie group G. In this paper, for each positive integer k , we associate an integer with fk,hi . We relate this number with Lefschetz coincidence number. We deduce that for any two differentiable maps f, there exists a positive integer k such that k 5.2+1 , and there is a point x C G such that ft (x) = (x) , where A is the rank of G . Introduction Let G be an n-dimensional com -pact connected Lie group with multip-lication p ( .e 44:0 xG--+G such that p ( x , y) = x.y ) and unit e . Let [G, G] be the set of homotopy classes of maps G G . Given two maps f , f G ---• Jollowing [3], we write f. f 'to denote the map G-.Gdefined by 01.11® =A/WO= fiat® ,sea Given a point g EC and a differ-entiable map F: G G , write GA to denote the tangent space of G at g [4,p.10] , and denote by d x F the linear map rig F :Tx0 T, (x)G induced by F , it is called the differential of Fat g [4,p.22]. Let LA, Rx :0 G be respec-tively the left translation Lx(i)=4..(g,e) , and the right translation Rx(1)./..(gcg). Then there is a natural homomorphism Ad ,the adjoin representation, from G to GL(G•), (the group of nonsingular linear transformations of Qdefined as follows:- Ad(g)= deRe, od,Lx. Note that d xRc, ad.; =d(4,( Lx(e)))0 de; =d.(4, 04)=4(40 Re) = d(4(4, (e)))0 (44, =d ar, o (44, . Since G is connected , the image of Ad belongs to the connected component of G(G)containing the identity,i.e. for each g E 0, detAd(g) > 0 . By Exercise Al • Dr.-Prof.-Department of Mathematics- College of Science- University of Baghdad. •• Dr.-Department of Mathematics- College of Science for Woman- University of Baghdad.


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