Slides of the plenary talk by Samuel Lomonaco - GW Links

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Free-groupsThus, the elements of F t F x form:wt j( )F x are of theThe Geometry of the Wirtinger Presentationw1wheret wj( ) tj( ) wj( )Note. This conjugation action is a leftaction.Wirtinger Generalized PresentationThe Wirtinger 3-D CW decomposition ofthe exterior X is nothing more than thenon-abelianfree resolution:^( , , ) FF1, 2,3^ F G F a b c r r r I^cbca r 1^acab r 2^babc r 3^ 2 1rrr1 2 3 ^1F^ Please note that and .IWirtinger Generalized Presentation^( , , ) FF1, 2,3We denote the above non-abelianfreeresolution more cryptically asabc , , : r, r, r:I1 2 3^ F G F a b c r r r Iand call it a generalized presentation.FWirtinger Generalized PresentationWirtinger Generalized Presentationabcd , , , : r1, r2, r3, r3:I^bcba r 1^cbcd r 2^daca r 3rrr ra c ca ca ^1 2 4 3 I^adbd r 46
Terminology for Generalized PresentationsThe current names identities, identities ofidentities, identities of identities ofidentities, etc. are too cumbersome. Sowe instead adopt the following terminology:generatorsor1-st orderrelators(2) (3) (4)x: r : r : r : 2-ndorderrelators3-rdorderrelators4-thorderrelatorsGeneralized Tietze MovesThere is only one generalized Tietze movefor each order, namely:n-thorder generalized Tietze move T n:( n) ( n1) ( n) ( n1) T : : r : r : : r : r :nwhereand where 1and F F r( n)Generalized Presentation EquivalenceThe definition of generalized presentationequivalence is a straight forward exercisefor the audience. So is the proof of thefollowing theorem:Theorem: Two generalized presentationsare equivalent iff there is a finite sequenceof generalized Tietze moves that transformone into the other.The Geometry of Generalized PresentationsProposition. Up to renaming and reordering, thereexists a one-toto-one one correspondence between theset of finite generalized presentations and a setof monopointedCW-complexes.x : r : r (3) : r (4) : Kx : r : r (3) : r(4): The generalized tietze moves correspond tosimple homotopy moves on the associated CW-complexes.Moreover, two generalized presentations defineCW complexes of the same simple homotopy typeiff there is a finite sequence of Tieze moveswhich transforms one into the other.The Geometry of the Fox Free Calculus(2) (3) ( n)Let P x: r : r : :r be a generalized presentation, and letLet the fundamental group of .Letassociated withK KPbe the corresponding CW-complex.1G K: F x Gx : r (2)Gbe the epimorphismFinally, let be the group ring of overthe integers .GKThe Geometry of the Fox Free CalculusLetKbe the universal cover of K, and letK KKerbe the non pathwise connectedspace above K.We now use the Fox free derivatives toconstruct a chain complex C C KHence,Moreover,* *H K H C* * *H *KH* G FC*7
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Slides of the plenary talk by Samuel Lomonaco - GW Links

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