Slides of the plenary talk by Samuel Lomonaco - GW Links

The Geometry of the Fox Free CalculusThe chain groups are defined as follows:• The 0-th chain groupis defined0 0 as the free FC C( )-module generated by the0-cell .• For n > 0 , the n-thchain groupnis defined as the free F -module( n)generated by the set of n-cells .nC C RR( n)The Geometry of the Fox Free CalculusThe boundary morphisms are defined asfollows:• For n = 0( )0 1C Cn = 0 ,Xxj1 Xj• For n > 0( n1) ( n)n > 0 , Cn1R CnR( n) rj R( n1)k rk R( n1) ( n)kk xWhere the Fox Free derivativesare geometrically defined as follows:/jKKerUniv. CoverCW-ComplexComplexCell Decompositions ofEpimorphism:KKKK, K,KF xG 1K(3) (4)w wXi wRj wRk wR4 (3) (4)g gXi gRj gRk gR4 X (3) (4)iRj RkR0-cells1-cells2-cells3-cellsg GwFx4-cells(2) (3) ( n)Recall x: r : r : : rand K K P.(2)If, for example, r rwith r x1x2x1x3,then the corresponding 2-cell Rin KisP x1xX2 1x X1 2x 2x1 R x 1x3X 1 x1x2x1X31 1R xx1 2X1 xX1 2 xxxX1 2 3r r r X X Xx x x1 2 31 2 3xx1 3(3)x1x3If, for example, urwith ur1 r2r1, thenthe boundary chain map of the corresponding 3-cell Uin Kis:U x r r x x r R rx Rx1 x3 x11 1 2 1 3 1 1 3 2^^u uR R r 1r 2 1 28