Some applications of coincidence theory (Q1345754)

The author considers a closed connected oriented \(n\)-manifold such that the rational singular cohomology \(H^* (M; \mathbb{Q})\) has a simple system of generators, i.e., there are \(x_1, \dots, x_\lambda \in H^* (M; \mathbb{Q})\) such that the graded \(\mathbb{Q}\)-vector space \(H^*(M; \mathbb{Q})\) has a basis consisting of \(1 \in H^0 (M; \mathbb{Q})\) and all monomials \(x_{j_1} \cup x_{j_2} \cup \cdots \cup x_{j_m}\) with \(1 \leq j_1 < \cdots < j_m \leq \lambda\). Let then \(f, g : M \to M\) be maps and reorder the generators \(x_1, \dots,x_\lambda\) as \(\alpha_1, \dots, \alpha_r\), \(\beta_1, \dots, \beta_s\) with \(r + s = \lambda\) such that the \(\alpha_i\) are of odd degree and the \(\beta_i\) are of even degree. For \(i = 1, \dots, r\) write \(f^* (\alpha_i) = \sum^r_{j = 1} a_{ji} \alpha_j + L_1\) where \(L_1\) contains the monomials of degree greater than one and similarly \(f^* (\beta_i) = \sum^s_{j = 1} \widehat {a}_{ji} \beta_j + L_1\), \(g^* (\alpha_i) = \sum^r_{j = 1} b_{ji} \alpha_j + L_1\) and \(g^* (\beta_i) = \sum^s_{j = 1} \widehat {b}_{ji} + L_1\). Denote by \(A\) and \(B\) the \((r,r)\)-matrices \((a_{ji})\) and \((b_{ij})\) and by \(\widehat {A}\) and \(\widehat {B}\) the \((s,s)\)-matrices \((\widehat {a}_{ij})\) and \((\widehat {b}_{ij})\). The author derives the following surprising formula for the Lefschetz coincidence number of \(f\) and \(g\): \(\Lambda(f, g) = \Phi (\widehat {B} + \widehat {A}) \text{det} (B - A)\) where, for a square \((n, n)\)-matrix \(C\), \(\Phi(C) = \sum c_{1\sigma(1)} \dots c_{n \sigma(n)}\) denotes the permanent of \(C\) (summation being over all permutations \(\sigma\) of \(\{1, \dots, n\})\). If \(M\) is in addition an \(H\)-manifold (i.e., there is an \(e \in M\) and a map \(m : M \times M \to M\) such that \(m(\cdot, e) = m(e,\cdot) = \text{id})\), let \(m_0 (x) = e\) and \(m_k(x) = m(x, m_{k - 1} (x))\). Let \(k - s > 1\). A primitive root of \(\text{Coinc} (m_k, m_s)\) is an \(x\in M\) such that \(m_k(x) = m_s(x)\) but \(m_i(x) \neq m_j(x)\) whenever \(k > i > j\), \(s \geq j \geq 0\) and \(k - j\) does not divide \(i - j\). Then \(\text{Coinc} (m_k, m_s)\) is partitioned into \(N\) totally primitive classes, i.e. classes all elements of which are primitive). The author precisely determines \(N = \text{ord} (\text{Coker } \mu_{k - s}) \prod (1 - q^{-r_\infty + r_q})\) where the product is over all prime divisors \(q\) of \(k - s\) and \(r_q\) denotes the number of cyclic subgroups of order a power of \(q\) of the fundamental group \(\pi_1 (M, e)\) and \(r_\infty\) the number of infinite cyclic subgroups \{this definition seems to have got lost in typesetting\} and \(\mu_{k - s}\) is multiplication by \(k - s\) in \(\pi_1(M,e)\).