By Vladimir Turaev

This ebook is an advent to combinatorial torsions of mobile areas and manifolds with detailed emphasis on torsions of three-d manifolds. the 1st chapters disguise algebraic foundations of the speculation of torsions and diverse topological buildings of torsions because of ok. Reidemeister, J.H.C. Whitehead, J. Milnor and the writer. We additionally speak about connections among the torsions and the Alexander polynomials of hyperlinks and 3-manifolds. The 3rd (and final) bankruptcy of the ebook bargains with so-called sophisticated torsions and the similar extra buildings on manifolds, in particular homological orientations and Euler buildings. As an software, we supply a development of the multivariable Conway polynomial of hyperlinks in homology 3-spheres. on the finish of the e-book, we in brief describe the new result of G. Meng, C.H. Taubes and the writer at the connections among the sophisticated torsions and the Seiberg-Witten invariant of 3-manifolds. The exposition is aimed toward scholars, expert mathematicians and physicists drawn to combinatorial points of topology and/or in low dimensional topology. the mandatory heritage for the reader comprises the easy fundamentals of topology and homological algebra.

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**Example text**

THE ORTHOGONALITY OF THE CHARACTERS = = 1 |G| 1 |G| tr (X (x)) tr X x−1 x∈G n m [X (x)]i,i X x−1 j,j . i=1 j=1 x∈G Let Pi,j be the n × m matrix with (k, l) -element equal to δi,k δl,j . Let Si,j = 1 |G| X (x) Pi,j X x−1 . x∈G Then Si,j i,j = = = 1 |G| 1 |G| 1 |G| [X (x)]i,k Pi,j k,l X x−1 l,j x∈G k,l [X (x)]i,k δi,k δl,j X x−1 l,j x∈G k,l [X (x)]i,i X x−1 j,j x∈G so n m Si,j χ, χ = i,j . 3, 0n×m 1 i,j n tr P Si,j = In if X X , if X ∼ X . But tr Pi,j = 0 unless i = j, in which case tr Pi,j = 1. Thus Si,j i,j 0 0 = 1 n if X X , if i = j, if i = j and X ∼ X .

Am,l xθ11 · · · xθrr ¯ r,l =il l=1 a1,l +···a il la xla1,l · · · xr r,l a1,l , . . , am,l n = xθ11 · · · xθrr xl1 + · · · + xln il = xθ11 · · · xθrr pα = [mθ ] pα , l=1 which completes the proof. To obtain the irreducible characters, we will use the following facts about the ring Λ of symmetric functions in the ground indeterminates x1 , x2 , . . For α ∈ Nn , let α aα = det xi j n×n . 1. FROBENIUS’S CONSTRUCTION Let δ = (n − 1, n − 2, . . , 1, 0) . Then (xj − xi ) , aδ = 1≤ij≤n the Vandermonde determinant.

I=1 j=1 x∈G Let Pi,j be the n × m matrix with (k, l) -element equal to δi,k δl,j . Let Si,j = 1 |G| X (x) Pi,j X x−1 . x∈G Then Si,j i,j = = = 1 |G| 1 |G| 1 |G| [X (x)]i,k Pi,j k,l X x−1 l,j x∈G k,l [X (x)]i,k δi,k δl,j X x−1 l,j x∈G k,l [X (x)]i,i X x−1 j,j x∈G so n m Si,j χ, χ = i,j . 3, 0n×m 1 i,j n tr P Si,j = In if X X , if X ∼ X . But tr Pi,j = 0 unless i = j, in which case tr Pi,j = 1. Thus Si,j i,j 0 0 = 1 n if X X , if i = j, if i = j and X ∼ X . 8 , whence χ, χ = 1 if X ∼ X , 0 if X X , and the result follows.