By Wolfgang Lück
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Additional resources for Algebraische Topologie
2) (Commutativity) For any two linear mappings T : E → E1 and S: E1 → E in Vf , we have tr(S ◦ T ) = tr(T ◦ S). 2) can be expressed in terms of commutative diagrams as follows. e. L = S ◦ T and L1 = T ◦ S, then tr(L) = tr(L1 ). 2. 2) Lemma. If the following diagram in Vf E T L1 L E / E1 T / E1 is commutative and T is an isomorphism, then tr(L) = tr(L1 ). Proof. Let S: E1 → E be the inverse of T . 2) we get tr(L) = tr(S ◦ L1 ◦ T ) = tr(L1 ◦ T ◦ S) = tr(L1 ) and the proof is completed. For given trace tr: Ef → K we deﬁne a function t: K → K by letting: t(a) = tr(L), where L: K → K is the endomorphism deﬁned by the formula L(x) = ax.
2g let f1 (αu ) = v=1 avu βv , where βj , 1 ≤ j ≤ 2h H (T Th ; Q). Let A = (aij ) be the matrix associated with f1∗ . are the generators of H 1 (T Observe that it can be written as: ⎛ A1,2 A3,4 . . A2g−1,2g ⎞ ⎜ ⎜ A=⎜ ⎝ 1,2 1,2 A1,2 3,4 .. A3,4 3,4 .. A1,2 2h−1,2h A3,4 2h−1,2h where A2l−1,2l 2m−1,2m = a2m−1,2l−1 a2m,2l−1 1,2 ⎟ ... A2g−1 3,4 ⎟ ⎟, .. ⎠ . 2g−1,2g . . A2h−1,2h a2m−1,2l a2m,2l . 1) Theorem. The Lefschetz coincidence number of the pair (f1 , f2 ) is given by: g h 2i−1,2i det(B2r−1,2r − A2i−1,2i 2r−1,2r ).
The ﬁrst result is about B(g, g, 1, 1). This means that the second map f2 : Sg → Sg has degree ±1, hence, up to homotopy, can be assumed to be a homeomorphism. By routine arguments, we know that the inclusion Coin(f1 , f2 ) → Fix(ff2−1 ◦ f1 ) is a homeomorphism which preserves the indices of the Nielsen classes, and |deg(f1 )| = 1. COINCIDENCE THEORY 35 |deg(ff2−1 ◦ f1 )|. So the coincidence question is equivalent to the ﬁxed point question. 12) Theorem. B(g, g, 1, 1) = 1 − χ(S Sg ) = 4g − 3. The proof of the above result contains two parts.
Algebraische Topologie by Wolfgang Lück