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| In mathematics, the '''Farrell–Jones conjecture''',<ref>Farrell, F.T., Jones, L.E., Isomorphism conjectures in algebraic K-theory, ''J. Amer. Math. Soc.'', v. 6, pp. 249–297, 1993</ref> named after [[F. Thomas Farrell]] (now at [http://www.math.binghamton.edu/farrell/ SUNY Binghamton]) and [[Lowell Edwin Jones]] (now at [http://www.math.sunysb.edu/~lejones/ SUNY Stony Brook]) states that certain [[assembly map]]s are [[isomorphism]]s. These maps are given as certain [[homomorphism]]s.
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| The motivation is the interest in the target of the assembly maps; this may be, for instance, the [[algebraic K-theory]] of a [[group ring]]
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| :<math>K_n(RG)</math>
| | They are typically a free website that are pre-designed for enabling businesses of every size in marking the presence on the internet and allows them in showcasing the product services and range through images, contents and various other elements. Affilo - Theme is the guaranteed mixing of wordpress theme that Mark Ling use for his internet marketing career. Step-4 Testing: It is the foremost important of your Plugin development process. After confirming the account, login with your username and password at Ad - Mob. The top 4 reasons to use Business Word - Press Themes for a business website are:. <br><br>If you are you looking for more in regards to [http://www.thekmichelle.com/forum/swift-solutions-worpress-some-insights-238836 wordpress backup plugin] review our own webpage. Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. You may either choose to link only to the top-level category pages or the ones that contain information about your products and services. We also help to integrate various plug-ins to expand the functionalities of the web application. This is identical to doing a research as in depth above, nevertheless you can see various statistical details like the number of downloads and when the template was not long ago updated. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Storing write-ups in advance would have to be neccessary with the auto blogs. Article Source: Stevens works in Internet and Network Marketing. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. Wordpress template is loaded with lots of prototype that unite graphic features and content area. <br><br>This advice is critical because you don't want to waste too expensive time establishing your Word - Press blog the exact method. By using Word - Press MLM websites or blogs, an online presence for you and your MLM company can be created swiftly and simply. Must being, it's beneficial because I don't know about you, but loading an old website on a mobile, having to scroll down, up, and sideways' I find links being clicked and bounced around like I'm on a freaking trampoline. Web developers and newbies alike will have the ability to extend your web site and fit other incredible functions with out having to spend more. Press CTRL and the numbers one to six to choose your option. |
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| or the [[L-theory]] of a group ring
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| :<math>L_n(RG)</math>,
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| where ''G'' is some [[group (mathematics)|group]].
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| The sources of the assembly maps are [[equivariant homology theory]] evaluated on the [[classifying space]] of ''G'' with respect to the family of [[virtually cyclic group|virtually cyclic subgroup]]s of ''G''. So assuming the Farrell–Jones conjecture is true, it is possible to restrict computations to virtually cyclic subgroups to get information on complicated objects such as <math>K_n(RG)</math> or <math>L_n(RG)</math>.
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| The [[Baum-Connes conjecture]] formulates a similar statement, for the [[topological K-theory]] of reduced group <math>C^*</math>-algebras <math>K^ {top}_n(C^r_*(G))</math>.
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| ==Formulation==
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| One can find for any ring <math> R</math> equivariant homology theories <math>KR^?_*,LR^?_*</math> satisfying
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| : <math>KR_n^G(\{\cdot\})\cong K_n(R[G])</math> respectively <math>LR_n^G(\{\cdot\})\cong L_n(R[G]).</math>
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| Here <math>R[G]</math> denotes the [[group ring]].
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| The K-theoretic Farrell–Jones conjecture for a group ''G'' states that the map <math>p:E_{VCYC}(G)\rightarrow \{\cdot\}</math> induces an isomorphism on homology
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| :<math>KR_*^G(p):KR_*^G(E_{VCYC}(G))\rightarrow KR_*^G(\{\cdot\})\cong K_*(R[G]).</math>
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| Here <math>E_{VCYC}(G)</math> denotes the [[classifying spaces for families|classifying space]] of the group ''G'' with respect to the family of virtually cyclic subgroups, i.e. a ''G''-CW-complex whose [[isotropy group]]s are virtually cyclic and for any virtually cyclic subgroup of ''G'' the [[fixed point set]] is [[contractible]].
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| The L-theoretic Farrell–Jones conjecture is analogous.
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| == Computational aspects ==
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| The computation of the algebraic K-groups and the L-groups of a group ring <math>R[G]</math> is motivated by obstructions living in those groups (see for example [[Wall's finiteness obstruction]], [[surgery obstruction]], [[Whitehead torsion]]). So suppose a group <math>G</math> satisfies the Farrell–Jones conjecture for algebraic K-theory. Suppose furthermore we have already found a model <math>X</math> for the classifying space for virtually cyclic subgroups: | |
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| : <math> \emptyset=X^{-1}\subset X^0\subset X^1\subset \ldots \subset X </math> | |
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| Choose <math>G</math>-pushouts and apply the Mayer-Vietoris sequence to them:
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| : <math> KR_n^G(\coprod_{j\in I_i} G/H_j\times S^{i-1})\rightarrow KR_n^G(\coprod_{j\in I_i} G/H_j\times D^i)\oplus KR_n^G(X^{i-1})\rightarrow KR_n^G(X^i) </math><math>\rightarrow KR_{n-1}^G(\coprod_{j\in I_i} G/H_j\times S^{i-1})\rightarrow KR_{n-1}^G(\coprod_{j\in I_i} G/H_j\times D^i)\oplus KR_{n-1}^G(X^{i-1}) </math>
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| This sequence simplifies to:
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| : <math> \bigoplus_{j\in I_i}K_n(R[H_j])\oplus \bigoplus_{j\in I_i} K_{n-1}(RH_j)\rightarrow \bigoplus_{j\in I_i} K_n(RH_j)\oplus KR_n^G(X^{i-1})\rightarrow KR_n^G(X^i) </math><math>\rightarrow \bigoplus_{j\in I_i}K_{n-1}(RH_j)\oplus\bigoplus_{j\in I_i}K_{n-2}(RH_j)\rightarrow \bigoplus_{j\in I_i} K_{n-1}(RH_j)\oplus KR^G_{n-1}(X^{i-1}) </math>
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| This means that if any group satisfies a certain isomorphism conjecture one can compute its algebraic K-theory (L-theory) only by knowing the algebraic K-Theory (L-Theory) of virtually cyclic groups and by knowing a suitable model for <math> E_{VCYC}(G)</math>.
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| ===Why the family of virtually cyclic subgroups ? ===
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| One might also try to take for example the family of finite subgroups into account. This family is much easier to handle. Consider the infinite cyclic group <math> \Z </math>. A model for <math>E_{FIN}(\Z)</math> is given by the real line <math>\R</math>, on which <math>\Z</math> acts freely by translations. Using the properties of equivariant K-theory we get
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| : <math>K_n^\Z(\R)=K_n(S^1)=K_n(pt)\oplus K_{n-1}(pt)=K_n(R)\oplus K_{n-1}(R).</math>
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| The [[Bass-Heller-Swan decomposition]] gives
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| : <math>K_n^\Z(pt)=K_n(R[\Z])\cong K_n(R)\oplus K_{n-1}(R)\oplus NK_n(R)\oplus NK_n(R).</math>
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| Indeed one checks that the assembly map is given by the canonical inclusion.
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| : <math>K_n(R)\oplus K_{n-1}(R)\hookrightarrow K_n(R)\oplus K_{n-1}(R)\oplus NK_n(R)\oplus NK_n(R)</math>
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| So it is an isomorphism if and only if <math>NK_n(R) =0</math>, which is the case if <math>R</math> is a [[regular ring]]. So in this case one can really use the family of finite subgroups. On the other hand this shows that the isomorphism conjecture for algebraic K-Theory and the family of finite subgroups is not true. One has to extend the conjecture to a larger family of subgroups which contains all the counterexamples. Currently no counterexamples for the Farrell–Jones conjecture are known. If there is a counterexample, one has to enlarge the family of subgroups to a larger family which contains that counterexample.
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| == Inheritances of isomorphism conjectures ==
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| The class of groups which satisfies the fibered Farrell–Jones conjecture contain the following groups
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| * virtually cyclic groups (definition)
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| * hyperbolic groups (see <ref>{{Citation | last1=Bartels | first1=Arthur | last2=Lück | first2 = Wolfgang | last3=Reich | first3=Holger| arxiv=math/0609685 | title=The K-theoretic Farrell-Jones Conjecture for hyperbolic groups| journal = Preprintreihe SFB 478 --- Geometrische Strukturen in der Mathematik | volume=434 | year= 2007 | place=Münster}}</ref>)
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| * CAT(0)-groups (see <ref>{{Citation | last1=Bartels | first1=Arthur | last2=Lück | first2 = Wolfgang | last3=Reich | first3=Holger| arxiv=0901.0442 | title=The Borel Conjecture for hyperbolic and CAT(0)-groups| journal = Preprintreihe SFB 478 --- Geometrische Strukturen in der Mathematik | volume=506 | year= 2009|place = Münster}}</ref>)
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| * solvable groups (see <ref>{{Citation | last1=Wegner | first1=Christian | arxiv=1308.2432 | title=The Farrell-Jones Conjecture for virtually solvable groups | year= 2013}}</ref>)
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| Furthermore the class has the following inheritance properties:
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| * closed under finite products of groups
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| * closed under taking subgroups.
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| == Meta-conjecture and fibered isomorphism conjectures ==
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| Fix an equivariant homology theory <math>H^?_*</math>. One could say, that a group '' G'' satisfies the isomorphism conjecture for a family of subgroups<math> F</math>, if and only if the map induced by the projection <math> E_F(G)\rightarrow \{\cdot\} </math> induces an isomorphism on homology:
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| : <math> H_*^G(E_F(G))\rightarrow H_*^G(\{\cdot\}) </math>
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| The group ''G'' satisfies the fibered isomorphism conjecture for the family of subgroups ''F'' if and only if for any group homomorphism <math> \alpha :H\rightarrow G</math> the group ''H'' satisfies the isomorphism conjecture for the family
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| : <math>\alpha^*F:=\{H'\le H|\alpha(H)\in F\}</math>.
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| One gets immediately that in this situation <math>H</math> also satisfies the fibered isomorphism conjecture for the family <math>\alpha^*F</math>.
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| ===Transitivity principle===
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| The transitivity principle is a tool to change the family of subgroups to consider. Given two families <math>F\subset F'</math> of subgroups of <math> G</math>. Suppose every group <math> H\in F'</math> satisfies the (fibered) isomorphism conjecture with respect to the family <math> F|_H:=\{H'\in F|H'\subset H\}</math>.
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| Then the group <math>G</math> satisfies the fibered isomorphism conjecture with respect to the family <math>F</math> if and only if it satisfies the (fibered) isomorphism conjecture with respect to the family <math>F'</math>.
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| ===Isomorphism conjectures and group homomorphisms===
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| Given any group homomorphism <math>\alpha:H\rightarrow G</math> and suppose that ''G"' satisfies the fibered isomorphism conjecture for a family ''F'' of subgroups. Then also ''H"' satisfies the fibered isomorphism conjecture for the family <math> \alpha^*F</math>. For example if <math> \alpha</math> has finite kernel the family <math> \alpha^*VCYC </math> agrees with the family of virtually cyclic subgroups of ''H''.
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| For suitable <math>\alpha</math> one can use the transitivity principle to reduce the family again.
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| == Connections to other conjectures==
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| ===Novikov conjecture ===
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| There are also connections from the Farrell–Jones conjecture to the [[Novikov conjecture]]. It is known that if one of the following maps
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| : <math>H^G_*(E_{VCYC}(G),L^{\langle-\infty\rangle}_R)\rightarrow H^G_*(\{\cdot\},L^{\langle-\infty\rangle}_R)= L^{\langle-\infty\rangle}_*(RG)</math>
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| : <math> H^G_*(E_{FIN}(G),K^{top}) \rightarrow H^G_*(\{\cdot\},K^{top}) = K_n(C^*_r(G))</math>
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| is rationally injective then the Novikov-conjecture holds for <math>G</math>. See for example,.<ref>Ranicki, A. A., "On the Novikov conjecture", ''In Novikov conjectures , index theorems and rigidity, Vol. 1'', (Oberwolfach 2003), pp. 272–337. Cambridge Univ. Press, Cambridge.</ref><ref>Lück, W. and Reich, H, "The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory", ''In Handbook of K-theory. Vol. 1,2'', pp. 703–842. Springer, Berlin, 2005.</ref> | |
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| ===Bost conjecture ===
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| The Bost conjecture states that the assembly map
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| :<math> H^G_*(E_{FIN}(G),K^{top}_{l^1})\rightarrow H^G_*(\{\cdot\},K^{top}_{l^1})=K_*(l^1(G)) </math>
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| is an isomorphism. The ring homomorphism <math> l^1(G)\rightarrow C_r(G)</math> induces maps in K-theory <math>K_*(l^1(G))\rightarrow K_*(C_r(G))</math>. Composing the upper assembly map with this homomorphism one gets exactly the assembly map occurring in the [[Baum-Connes conjecture]].
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| :<math> H^G_*(E_{FIN}(G),K^{top}_{l^1})=H^G_*(E_{FIN}(G),K^{top})\rightarrow H^G_*(\{\cdot\},K^{top})=K_*(C_r(G))</math>
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| ===Kaplansky conjecture ===
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| The Kaplansky conjecture predicts that for an integral domain <math>R</math> and a torsionfree group <math>G</math> the only idempotents in <math>R[G]</math> are <math>0,1</math>. Each such idempotent <math>p</math> gives a projective <math>R[G]</math> module by taking the image of the right multiplication with <math>p</math>. Hence there seems to be a connection between the Kaplansky conjecture and the vanishing of <math>K_0(R[G])</math>. There are theorems relating the [[Kaplansky conjecture]] to the Farrell–Jones conjecture (compare <ref>{{Citation | doi=10.1112/jtopol/jtm008 | last1=Bartels | first1=Arthur | last2=Lück | first2=Wolfgang | last3= Reich | first3=Holger | title=On the Farrell-Jones Conjecture and its applications | arxiv=math/0703548| journal=Journal of Topology | volume=1 | year=2008 | issue=1 | pages = 57–86}}</ref>).
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Farrell-Jones Conjecture}}
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| [[Category:Surgery theory]]
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| [[Category:K-theory]]
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| [[Category:Conjectures]]
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