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# THE FAMILY SOC 352

UW

GPA 3.8

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This 86 page Class Notes was uploaded by Zelda Parker on Wednesday September 9, 2015. The Class Notes belongs to SOC 352 at University of Washington taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/192046/soc-352-university-of-washington in Sociology at University of Washington.

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Date Created: 09/09/15

A Lemma of CTC Wall B ra unschweig Septem ber 2007 Graham Ellis httpwwwmathsnuigalwayie Suppotrded by Marie Curie MTKD CT 2006 O42685 Uquot gt 39 Contents Definitions Example computations Contracting homotopies The lemma Potential computations 1 DEFINITIONS k integral domain k integral domain A kx17 7Xnl an associative algebra k integral domain A kx17 7Xnl an associative algebra A sequence of A module homomorphisms d d d d AR33gtR22gtR11gtR0 is said to be a free A resolution of M if k integral domain A kx17 7Xnl an associative algebra A sequence of A module homomorphisms d d d d AR33gtR22gtR11gtR0 is said to be a free A resolution of M if gt Exactness kerd image dn1 for all n 2 1 k integral domain A kx17 7Xnl an associative algebra A sequence of A module homomorphisms d d d d AR33gtR22gtR11gtR0 is said to be a free A resolution of M if gt Exactness kerd image dn1 for all n 2 1 gt Freeness Rn is a free A module for all n 2 0 k integral domain A kx17 7Xnl an associative algebra A sequence of A module homomorphisms d d d d AR33gtR22gtR11gtR0 is said to be a free A resolution of M if gt Exactness kerd image dn1 for all n 2 1 gt Freeness Rn is a free A module for all n 2 0 gt Augmentation the cokernel of d1 is isomorphic to the module M For A modules M N define kerHomARn7 N a HomARn17 N EXtAUVL N imageHomARn17 N a HomARn7 N For A modules M N define kerHomARn7 N a HomARn17 N EXtAUVL N imageHomARn17 N a HomARn7 N and kerRn A N H Rnil A N A 7 Tor M7 N 7 imageRn1 A N a Rn A N There are many reasons for wanting to calculate these functors My motivation is not A system is controllable if one can move from one system trajectory X0 to another trajectory X1 without violating the system law Some systems are more controllable than others There are many reasons for wanting to calculate these functors My motivation is not A system is controllable if one can move from one system trajectory X0 to another trajectory X1 without violating the system law Some systems are more controllable than others For an A module M arising from a system set N HomAM7 A Definition The controllability degree of M is the first natural number n gt 0 such that Ext NA 7 O and Ext MN7 A O for O lt i lt n The are a number of packages for computing these functors COCOA MACAULAY and SINGULAR contain a range of Grobner basis methods for computing the functors Torf M7 N and Ext M7 N in the case where k is a field and the ring A is commutative The are a number of packages for computing these functors COCOA MACAULAY and SINGULAR contain a range of Grobner basis methods for computing the functors Torf M7 N and Ext M7 N in the case where k is a field and the ring A is commutative The PLURAL extension to Singular handles certain non commutative rings A For the cohomology of a group G one takes the ring of integers k Z the module M Z with trivial G action the group ring A ZG and sets H G N Extgdl N HnG N TornZGZ N For the cohomology of a group G one takes the ring of integers k Z the module M Z with trivial G action the group ring A ZG and sets H G N Extgdl N HnG N TornZGZ N CAP and MAGMA handle n 12 MAGMA handles n gt 2 for G a small p group where it suffices to set k GFp For the cohomology of a group G one takes the ring of integers k Z the module M Z with trivial G action the group ring A ZG and sets H G N Extgdl N HnG N TornZGZ N CAP and MAGMA handle n 12 MAGMA handles n gt 2 for G a small p group where it suffices to set k GFp The computation of cohomology involves two expensive but independent tasks 1 the computation of a free resolution 2 the computation of the homology of a chain complex This talk focuses on a method for task 1 2 EXAMPLE COMPUTATIONS Theorem The group K3 kerSL2Z33 a SL2Zg has third integral homology group of exponent 27 ii In dimensions n 7 37 1 g n it has integral homology HnK3Z ofexponent at most 9 Proof i W Browder and J Pakianathan Cohomology of uniformly powerful p groups Trans Amer Math Soc 352 2000 no 6 2659 2688 ii J Pakianathan Exponents and the cohomology of finite groups Proc Amer Math Soc 128 2000 no 7 1893 1897 Theorem The group K3 kerSL2Z33 a SL2Zg has third integral homology group of exponent 27 ii In dimensions n 7 37 1 g n g 6 it has integral homology HnK3Z ofexponent at most 9 Automated Proof gapgt K3MaximalSubgroupsSylowSubgroup SL2Integers mod 3398 3 2 gapgt K3ImageIsomorphismPcGroupK3 gapgt DisplayList 1 4 n gtGroupHomologyK3 n 3 3 3 3 3 3 J 3 3 3 3 3 3 27 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 9 9 9 3 3 3 3 3 3 3 3 3 3 9 9 9 9 9 Theorem The Mathieu group M23 has trivial integral homology HrM237 Z 0 in dimensions n 17 27 3 Proof RJ Milgram The cohomology of the Mathieu group M23 J Group Theoty3 2000 no 1 7 26 Theorem The Mathieu group M23 has trivial integral homology HrM237 Z O in dimensions n 17 27 3 Automated Proof gapgt GroupHomologyMathieuGroup23 1 gapgt GroupHomologyMathieuGroup23 2 gapgt GroupHomologyMathieuGroup23 3 Theorem The mod2 cohomology H M11Zg of the Mathieu group M11 is a vector space of dimension equal to the coef cients ofx in the Poincare series X47X3X27X1X67X5X472X3X27X1 for all n Proof PJ Webb A local method in group cohomologyquot Comment Math Helv 62 1987 no 1 135 167 Theorem The mod2 cohomology H M11Zg of the Mathieu group M11 is a vector space of dimension equal to the coef cients ofx in the Poincare series X47X3X27X1X67X5X472X3X27X1 for all n g 20 Automated Proof gapgt PoincareSeriesPrimePartMathieuGroup11 220 xquot4xquot3xquot2x1xquot6xquot5xquot42xquot3xquot2x1 3 CONTRACTING HOMOTOPIES A free ZG resolution d HRnQRn71gtgtR0 is represented in HAP as a component object A free ZG resolution dn HRnHRn71gtHR0 is represented in HAP as a component object Rgroup the group G A free ZG resolution HRniRn1HHR0 is represented in HAP as a component object Rgroup the group G Rets a partial listing of elements of G A free ZG resolution HRniRn1HHR0 is represented in HAP as a component object Rgroup the group G Relts a partial listing of elements of G Rdimensionn Rankzan A free ZG resolution sRniRn1ssR0 is represented in HAP as a component object Rgroup the group G Relts a partial listing of elements of G Rdimensionn Rankzan Rboundarynk a list of integer pairs representing the boundary of the kth free generator in degree n A free ZG resolution sRniRn1ssR0 is represented in HAP as a component object Rgroup the group G Relts a partial listing of elements of G Rdimensionn Rankzan Rboundarynk a list of integer pairs representing the boundary of the kth free generator in degree n And more The following element of choice occurs frequently in homological algebra For each X E kerdn Rn a Rn71 choose an element quotlt E Rn1 such that dn1quotlt X The following element of choice occurs frequently in homological algebra For each X E kerdn Rn a Rnsl choose an element quotlt E Rn1 such that dn1quotlt X This choice is made using the resolution component Rhomotopynig contracting homotopy The following element of choice occurs frequently in homological algebra For each X E kerdn Rn a Rnsl choose an element X E Rn1 such that dn1X X This choice is made using the resolution component Rhomotopynig contracting homotopy If G is large or infinite one can39t solve dn1X X using basic linear algebra over Z A contracting homotopy on RE is a family of abelian group homomorphisms hn Rn a Rn n 2 0 satisfying dn1hnx i hnildnx X for all X E Rn where h1 0 A contracting homotopy on RE is a family of abelian group homomorphisms hn Rn a Rn n 2 0 satisfying dn1hnx i hnildnx X for all X E Rn where h1 0 Since the hn are not G equivariant one needs to specify hnx on a set of abelian group generators for Rn A contracting homotopy on RE is a family of abelian group homomorphisms hn Rn a Rn n 2 0 satisfying dn1hnx i hnildnx X for all X E Rn where h1 0 Since the hn are not G equivariant one needs to specify hnx on a set of abelian group generators for Rn Lemma Setting hnx ensures dn1quotlt X So we need a range of methods for providing contracting homotopies Geometry can provide resolutions with contracting homotopy Geometry can provide resolutions with contracting homotopy Example G ltXy X7 y 1 acts freely on the contractible space X R2 Geometry can provide resolutions with contracting homotopy Example G ltXy X7 y 1 acts freely on the contractible space X R2 There is a G equivariant ceuar decomposition of X R2 xyxE Geometry can provide resolutions with contracting homotopy Example G ltXy X7 y 1 acts freely on the contractible space X R2 There is a G equivariant cellular decomposition of X R2 CnX free abeian group on n cells X contractible cellular chain complex CX is exact cm a c300 cm i Clix Com GltXyxylgt xyxE xny 3quot VV yE E F xE V xV E xE We View mmmagm qmqm gm as a chain complex of ZG modules H0ZGZG ZGZG GltXyxylgt xyxE xny 3quot VV yE E F xE V xV E xE We View mmmagm qmqm gm as a chain complex of ZG modules H0ZGZG ZGZG d2F 17XE y 71E A homotopy homomorphism ho C0X a C1X can be specified by setting Y0 V and choosing a maximal contractible cellular subspace Y1 in the 1 skeleton YV XVV A homotopy homomorphism ho C0X a C1X can be specified by setting Y0 V and choosing a maximal contractible cellular subspace Y1 in the 1 skeleton YV XVV Now for example h0xyV E l yE since this corresponds to the path from V to xyV in Y1 A homotopy homomorphism ho C0X a C1X can be specified by setting Y0 V and choosing a maximal contractible cellular subspace Y1 in the 1 skeleton Now for example h0xyV E l yE since this corresponds to the path from V to xyV in Y1 And h0V 0 since V 6 Y0 A homotopy homomorphism hl C1X a C2X can be specified by choosing a maximal contractible cellular subspace Y1 in the 1 skeleton and a maximal contractible cellular subspace Y2 of the 2 skeleton l xyxE kW xyE E F V xV E A homotopy homomorphism hl C1X a C2X can be specified by choosing a maximal contractible cellular subspace Y1 in the 1 skeleton and a maximal contractible cellular subspace Y2 of the 2 skeleton l xyxE v y E yV E xyE E F V xV E Now for example h1xyE 0 since xyE is in Y1 A homotopy homomorphism hl C1X a C2X can be specified by choosing a maximal contractible cellular subspace Y1 in the 1 skeleton and a maximal contractible cellular subspace Y2 of the 2 skeleton l xyxE v y E yV E xyE E F V xV E Now for example h1xyE 0 since xyE is in Y1 And h1xyXE yF l xyF since this is the path from Y1 to xyXE in Y2 A homotopy homomorphism hl C1X a C2X can be specified by choosing a maximal contractible cellular subspace Y1 in the 1 skeleton and a the maximal contractible cellular subspace of the 2 skeleton E I I Now for example h1xyE 0 since xyE is in Y1 And h1xyXE yF l xyF since this is the path from Y1 to xyXE in Y2 4 THE LEMMA Let A be a ring Let Ct HCnHCn71gtgtC0 be an A resolution of some A module M where the A modules C are NOT assumed to be free Let A be a ring Let Ct HCnHCn71gtgtC0 be an A resolution of some A module M where the A modules C are NOT assumed to be free Suppose that for each m we have a free A resolution of Cm Dmti gt 0m gt Dmn1 gt gt Dmp gt Cm Let A be a ring Let CL HCnHCn71gtgtC0 be an A resolution of some A module M where the A modules C are NOT assumed to be free Suppose that for each m we have a free A resolution of Cm Dmat gt 0m gt Dmn1 gt gt Dmp gt Cm Lemma CTC Wall There exists a free A resolution RE A M With Rn EB 0m pqn Let A be a ring Let CL HCnHCn71gtgtC0 be an A resolution of some A module M where the A modules C are NOT assumed to be free Suppose that for each m we have a free A resolution of Cm Dmci gt 0m gt Dmn1 gt gt Dmp gt Cm Lemma CTC Wall There exists a free A resolution RE A M With Rn EB 0m pqn The proof can be made constructive using the notion of contracting homotopy Furthermore one can derive an explicit formula for a contracting homotopy on Rt in terms of contracting homotopies on the Dmc and CL MM 032 DQQ 012 D 02 i it i i 031 021 D11 001 do 030 020 D10 000 iiii HC HCZHClHCO d d 0 HM 6032902260129002 i gtDs1gt021 gt011 gt001 lidoii d1 gt030gt020gtD10gtDoo i nagagaQHCo HM 6032902260129002 i gtDs1gt021 gt011 gt001 lidoii d1 gt030gt020gtD10gtDoo i nagagaQHCo 8d0d1 HM 6032902260129002 i gtDs1gt021 gt011 gt001 lidoii d1 gt030gt020gtD10gtDoo i nagagaQHCo 8d0d1 but for dld1 7 O 6032902260129002 gtDs1gt021 gt011 gt001 d do d1 gt030gt020gtD10gtDoo i nagagaQHCo 6032902260129002 gtDs1gt021 gt011 gt001 d do d1 gt030gt020gtD10gtDoo i nagagaQHCo 6vd dld2 6032902260129002 gtDs1gt021 gt011 gt001 d do d1 gt030gt020gtD10gtDoo i nagagaQHCo 6vd dld2 but for dzd2 7 0 etc Lemma CTC Wall There is a free A resolution R a M With Rn EB om pqn and boundary homomorphism 8d d1d2d3m On any summand Dp all but finitely many di are zero 5 POTENTIAL COMPUTATIONS SCENARIO 1 Let N be a normal subgroup of C Set Q GN Let Cat be a free ZQ resolution of Z We can produce suitable resolutions Dmat from a free ZN resolution of Z Wall39s lemma was proved in this context and provides a free ZG resolution of Z SCENARIO 1 Let N be a normal subgroup of C Set Q GN Let C be a free ZQ resolution of Z We can produce suitable resolutions Dmc from a free ZN resolution of Z Wall39s lemma was proved in this context and provides a free ZG resolution of Z This technique underlies HAP functions such as ResolutionNilpotentGroup Gn ResolutionSubnormalSeries GN1 N2 Nk n Proposition The free nilpotent group G of class two on 4 generators has integral cohomology groups H1GZZ4 H2GZZzo H3GZZ56 H4GZZg4 H5GZZ ZQO H6GZZ Zg4 H7GZBZ56 H8Gzg220 H9Gzgz4 H10GZ z H GZ 0 n 11 The ring HGZ is generated by 4 classes in degree 1 20 classes in degree 2 36 classes in degree 3 and 20 classes in degree 4 Proposition The free nilpotent group G of class two on 4 generators has integral cohomology groups H1GZZ4 H2GZZzo H3GZZ56 H4GZZg4 H5GZZ ZQO H6GZZ Zg4 H7GZBZ56 H8Gzg220 H9Gzgz4 H10GZ z H GZ 0 n 11 The ring HGZ is generated by 4 classes in degree 1 20 classes in degree 2 36 classes in degree 3 and 20 classes in degree 4 The additive structure of HG7 Z was first calculated in Larry Lambe Cohomology of principal G bundles over a torus when HBGR is polynomial Bulletin Soc Math de Belgium 38 1986 247 264 gapgt n3 m7 gapgt FFreeGroup3 GNilpotentQuotientF2 gapgt RResolutionNilpotentGroupG10 gapgt for n in 1m1 do gt Print Cohomology in dimension n gt CohomologyHomToIntegersR n n od Cohomology in dimension 1 0 0 0 0 0 0 0 0 0 0 Cohomology in dimension Cohomology in dimension 0 0 0 Cohomology in dimension 4 0 0 0 0 0 0 0 0 Cohomology in dimension 5 0 0 0 Cohomology in dimension 6 0 gapgt DimensionR 7 0 gapgtList 1 ml n gtLengthIntegralRingGeneratorsRn 3 8 6 0 0 0 0 000 0000 20 30 gapgt FFreeGroup4 GNilpotentQuotientF2 gapgt LGLowerCentralSeriesLieAlgebraG gapgt LieAlgebraHomologyLG8 00 0 00 0 0 00 0 0 00 0 00 0000 gapgt FFreeGroup4 GNilpotentQuotientF2 gapgt LGLowerCentralSeriesLieAlgebraG gapgt LieAlgebraHomologyLG8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Compare gapgt GroupHomologyltG 8 0000000000000000 0000 SCENARIO 2 Let G be the fundamental group of a graph of groups For instance an amalgamated free product G PA Q corresponding to the graph P Q 0 A Let C be the cellular chain complex of the graph We can produce suitable resolutions Dmc from free ZGe resolutions for the edge groups Ge and free ZGV resolutions for the vertex groups GV SCENARIO 2 Let G be the fundamental group of a graph of groups For instance an amalgamated free product G PA Q corresponding to the graph P Q 0 A Let C be the cellular chain complex of the graph We can produce suitable resolutions Dmc from free ZGe resolutions for the edge groups Ge and free ZGV resolutions for the vertex groups GV This technique underlies the HAP function ResolutionGraphOfGroupsGn The amalgamated free product 55 53 54 can be represented as a graph of groups gapgt SSSymmetricGroup5 gapgt S4SymmetricGroup4 gapgt SSSymmetricGroup3 gapgt 8385 GroupHomomorphismByFunction83 SS x gtx gapgt 8384 GroupHomomorphismByFunction83 S4 x gtx gapgt DSSS4 5355535411 gapgt RResolutionGraphOfGroupsD8 gapgt HomologyTensorWithIntegersR 7 22 2460 SO H755 53 547 Z Zz3 69 Z4 69 Z60 SCENARIO 3 Let G act on some cellular contractible space X such that cells are permuted Let Ct CX We can produce suitable resolutions Dmt from free ZGe resolutions for the stabilizer groups Ge G of cells e Cellular space X can sometimes be produced using POLYMAKE software Orbit Polytopes Let 042 G a GKR be a faithful representation Let v E R have trivial stabilizer group Definition PG Convex hull agv g E G G A4 acts on v X17X27X37X4 E R4 by adv XgilawXg127Xg137Xg714 G A4 acts on v X17X27X37X4 6R4 by adv walwxrlawXgiwswxriw For v 172374 we get G A4 acts on v X17X27X37X4 61 by agv XgilmvXg7127Xg1sng14 For v 172374 we get X 132y 1234z 243 The HAP function PolytopalComplexG v n uses POLYMAKE to produce n terms of the non free resolution CX and compute the stabilizer subgroups Ge Wall39s lemma not yet implemented for this case The HAP function PolytopalComplexG v n uses POLYMAKE to produce n terms of the non free resolution CX and compute the stabilizer subgroups Ge Wall39s lemma not yet implemented for this case But we can still use CX to find a presentation of G gapgt GSylowSubgroupAlternatingGroup18 3 gapgt PPolytopa1ComplexG123456789 gt1011121314151617182 gapgt PresentationOfResolutionP rec freeGroup ltfree group on the generators f1 f2 f3 f4 f5 f6 f7 f8 gt relators f1quot3 f2f1f2quot 1f1quot1 f3f1f3quot 1f1quot1 f4f3f4quot 1f1quot1 f1f4quot1f2quot1f4 f5f1f5quot 1f1quot1 f6f1f6quot 1f1quot1 f7f1f7quot 1f1quot1 f8f1f8quot 1f1quot1 f2quot3 f3f2f3quot 1f2quot1 f2f4quot1f3quot1f4 f5f2f5quot 1f2quot1 f6f2f6quot 1f2quot1 f7f2f7quot 1f2quot1 f8f2f8quot 1f2quot1 f3quot3 f5f3f5quot 1f3quot1 f6f3f6quot 1f3quot1 f7f3f7quot 1f3quot1 f8f3f8quot 1f3quot1 f4quot3 f5f4f5quot 1f4quot1 f6f4f6quot 1f4quot1 f7f4f7quot 1f4quot1 f8f4f8quot 1f4quot1 f5quot3 f6f5f6quot 1f5quot1 f7f5f7quot 1f5quot1 f8f7f8quot 1f5quot1 f5f8quot1f6quot1f8 f6quot3 f7f6f7quot 1f6quot1 f6f8quot1f7quot1f8 f7quot3 f8quot3 POSSIBLE SCENARIO 4 Resolutions in commutative algebra

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