By Stefan Jackowski, Bob Oliver, Krzysztof Pawalowski

ISBN-10: 3540540989

ISBN-13: 9783540540984

As a part of the clinical job in reference to the seventieth birthday of the Adam Mickiewicz collage in Poznan, a world convention on algebraic topology was once held. within the ensuing court cases quantity, the emphasis is on enormous survey papers, a few offered on the convention, a few written in this case.

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**Example text**

15 Theorem (Waldhausen). Two knots k1 , k2 in S 3 with the peripheral systems (Gi , mi , li ), i = 1, 2, are equal if there is an isomorphism ϕ : G1 → G2 with the property that ϕ(m1 ) = m2 and ϕ(l1 ) = l2 . Proof. 6], the isomorphism ϕ is induced by a homeomorphism h : C1 → C2 mapping representative curves µ1 , λ1 of m1 , l1 onto representatives µ2 , λ2 of m2 , l2 . The representatives can be taken on the boundaries ∂C i . 27. 14. Together h and h define the required homeomorphism h : S 3 → S 3 which maps the (directed) knot k1 onto the (directed) k2 .

This implies that ϕ(ua ) = (u a )ε for ε ∈ {1, −1}. Now, u a (u c v d )ab = ϕ(ua (uc v d )−ab ) = ϕ(ua )ϕ(uc v d ))−ab = (u a )ε (u c v d )−ab ; hence, (u a )1−ε = (u c t d )−2ab . This equation is impossible: the homomorphism G∗ → G∗ /Z∗ ∼ = Za ∗ Zb maps the term on the left onto unity, whereas the term on the right represents a non-trivial element of Za ∗ Zb because a c and b d . 3]. F Asphericity of the Knot Complement In this section we use some notions and deeper results from algebraic topology, in particular, the notion of a K(π, 1)-space, π a group: X is called a K(π, 1)-space if π1 X = π and πn X = 0 for n = 1.

There is a relation between the genera of a knot and its companion. 10 Proposition (Schubert). 8). Denote by g, ˆ g, g, ˜ the genera of k, number of k and a meridian m ˆ of a tubular neighbourhood Vˆ of kˆ which contains k. Then g ≥ ng˜ + g. ˆ This result is due to H. Schubert [1953]. 11 Lemma. There is a Seifert surface S of minimal genus g spanning the satellite ˆ k such that S ∩ ∂ Vˆ consists of n homologous (on ∂ Vˆ ) longitudes of the companion k. 3 ˆ The intersection S ∩ (S − V ) consists of n components.

### Algebraic Topology, Poznan 1989 by Stefan Jackowski, Bob Oliver, Krzysztof Pawalowski

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