By L. Smith

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Extra resources for Algebraic Topology, Gottingen 1984

Sample text

Finally given a link L we also drop the representation from the notation when the representation is the trivial representation to GL(1, Z). With all these conventions, given a knot K ⊂ S 3 , the polynomial Z[t ±1 ] is just the ordinary Alexander polynomial. 5 Computation of Twisted Alexander Polynomials Let N be a 3-manifold with empty or toroidal boundary, α : π1 (N ) → GL(k, R[F ]) a representation with R a Noetherian UFD and F a free abelian group. Given a finite presentation for π1 (N ) the polynomials αN,1 ∈ R[F ] and αN,0 ∈ R[F ] can be computed efficiently using Fox calculus (cf.

In this section we will give an algorithm for computing τ (N, γ ⊗ ψ) which is based on ideas of Turaev (cf. 2]). We will first consider the case that N is closed. We write π = π1 (N ). We endow N with a CW-structure with one 0-cell, n 1-cells, n 2-cells and one 3-cell. It is wellknown that such a CW-structure exists (cf. g. 1]). Using this CW-structure we have the cellular chain complex ∂3 ∂2 ∂1 → C2 (N˜ ) − → C1 (N˜ ) − → C0 (N˜ ) → 0 0 → C3 (N˜ ) − ∼ Z[π]n for i = 1, 2. Let Ai , i = ˜ ∼ where Ci (N) = Z[π] for i = 0, 3 and Ci (N˜ ) = 1, 2, 3 be the matrices over Z[π] corresponding to the boundary maps ∂i : Ci → Ci−1 with respect to the bases given by the lifts of the cells of N to N˜ .

Theorem 6 Let i : X n → M n+2 be a PL embedding of an oriented compact pseudomanifold in an oriented compact manifold such that the pair (M, X) is stratifiable without odd-codimensional strata. Assume that all Poincaré local systems BVR are strongly transverse to the singular set V − V , V ∈ X . Then L∗ (X) = L∗ (M, X) − V ∈X iV ∗ (ch[BVR ]K ∩ L∗ (V )). 9) holds automatically if X happens to be a manifold and the singular set of the embedded X has codimension at least 3. For in that case, the links in X are spheres of dimension 2 or higher which are simply connected.