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Fixed points and coincidences in torus bundles. (English) Zbl 1226.55002

This is an extremely interesting article studying Nielsen and Reidemeister theory for fixed and coincidence points for fiberwise maps of torus bundles using bordism theory. For a thorough understanding one should first work through the article [Topol. Methods Nonlinear Anal. 33, No. 1, 85–103 (2009; Zbl 1178.55002)] by D. L. Gonçalves and the author.
Let \(p_M:M\to B\) and \(p_N:N\to B\) be smooth fiber bundles over \(B\) with fiber dimension \(m\) and \(n\), resp., and \(b:=\dim B\). Let \(f_1,f_2:M\to N\) be fiberwise maps. Denote by \(P(N)\) the space of all continuous paths \(\theta:[0,1]\to N\) and denote by \(E_B(f_1,f_2)\) the space of pairs \((x,\theta)\in M\times P(N)\) such that \(p_N\circ\theta=p_M(x)\), where \(\theta\) is a path joining \(f_1\) to \(f_2\). Call \(pr:E_B(f_1,f_2)\to M\) the projection. Define a virtual vector bundle \(\phi:=f_1^*(TN)-TM-p_M^*(TB)\) and let \(\tilde{\phi}:=pr^*(\phi)\). In the article mentioned above, the authors constructed a normal bordism class \(\tilde{\omega}_B(f_1,f_2)\in\Omega_{m+b-n}(E_B(f_1,F_2);\tilde{\phi})\).
A path component \(Q\) of \(E_B(f_1,f_2)\) is said to be essential if \(\tilde{\omega}_B(f_1,f_2)\) restricts to a nontrivial bordism class in \(\Omega_*(Q;\tilde{\phi}|\,Q)\). The Nielsen number \(N_B(f_1,f_2)\) is the number of essential path components of \(E_B(f_1,f_2)\) and the geometric Reidemeister number \(R_B(f_1,f_2)\) is the set \(\pi_0(E_B(f_1,f_2))\) of all path components of \(E_B(f_1,f_2)\). Denote by \(MC_B(f_1,f_2)\) the minimum number of coincidence points among all pairs \((f^\prime_1,f^\prime_2)\) such that \(f^\prime_1\) and \(f^\prime_2\) are fiberwise homotopic to \(f_1\) and \(f_2\). Finally denote by \(MCC_B(f_1,f_2)\) the minimum number of path components of coincidence subspaces of \(M\) among all pairs which are fiberwise homotopic to \((f_1,f_2)\). One then has that \(N_B(f_1,f_2)\leq MCC_B(f_1,f_2)\leq MC_B(f_1,f_2)\).
The author considers the case where \(p_M:M\to B\) and \(p_N:N\to B\) are linear torus bundles (i.e., the typical fiber is a torus and the structure group is \(\text{GL}(m,\mathbb{Z})\) which acts on \((\mathbb{R}^m,\mathbb{Z}^m)\) and hence on the torus \(T^m\)) and \(B\) is either a point or a sphere. He then shows that \(MCC_B(f_1,f_2)=N_B(f_1,f_2)\) and that \(MC_B(f_1,f_2)=N_B(f_1,f_2)\) if \(N_B(f_1,f_2)=0\) or \(m+b=n\) and \(\infty\) else. Moreover, he explicitly computes the Reidemeister numbers in this situation.

MSC:

55M20 Fixed points and coincidences in algebraic topology
55R10 Fiber bundles in algebraic topology
55S35 Obstruction theory in algebraic topology

Citations:

Zbl 1178.55002
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References:

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