Kervaire invariants and selfcoincidences (Q1954318)

Let \(M\) and \(N\) be manifolds and \(f_1,f_2:M\to N\) continuous. Denote the coincidence set by \(C(f_1,f_2):=\{x\in M|\;f_1(x)=f_2(x)\). One wants to determine the numbers \(\text{MC}(f_1,f_2):=\min\{\#C(f'_1,f'_2)|\;f'_1\sim f_1, f'_2\sim f_2\}\) and \(\text{MCC}(f_1,f_2):=\min\{\#\pi_0(C(f'_1,f'_2))|\;f'_1\sim f_1, f'_2\sim f_2\}\). Finally, denote by \(N^\#(f_1,f_2)\) the Nielsen number. The pair \((f_1,f_2)\) is called loose if there are homotopies \(f'_1\sim f_1\) and \(f'_2\sim f_2\) such that \(f'_1\) and \(f'_2\) are coincidence free. For a map \(f:M\to M\) the pair \((f,f)\) is said to be loose by small deformation if for every metric on \(N\) and every \(\epsilon>0\) there is an \(\epsilon\)-approximation \(f'\) of \(f\) such that \(f\) and \(f'\) are coincidence free. The authors consider the case where \(M=S^m\) and \(N\) is a spherical space form, i.e., \(N=S^n/G\) where \(G\) is a finite group acting smoothly and freely on \(S^n\). The authors deal with the question of when \(\text{MCC}(f_1,f_2)\) is different from \(N^\#(f_1,f_2)\) and when \((f,f)\) is loose but not loose by small deformation. As to the first question, they show that a necessary condition for \(\text{MCC}(f_1,f_2)\not= N^\#(f_1,f_2)\) is that \(f_1\) and \(f_2\) be homotopic. As for the second question, let \([f]\in\pi_m(N)\). Then \(\text{MC}(f,f)=\text{MCC}(f,f)\) and these numbers as well as \(N^\#(f,f)\) can take only 0 and 1 as possible values. If, in addition, if \([f]\) lifts to \([\tilde{f}]\in\pi_m(S^n)\) denote by \(V_{n+1,2}:=ST(S^n)\) the Stiefel manifold of unit tangent vectors to \(S^n\) fibered over \(S^n\). Consider the homotopy sequence \[ \dots@>>> \pi_m(V_{n+1,2} @>>> \pi_m(S^n) @>\partial>>\pi_m(S^{n-1}) @>\text{incl}>>\pi_{m-1}(S^{n-1}) @>>> \dots \] and denote by \(E:\pi_{m-1}(S^{n-1})\to\pi_m(S^m)\) the suspension homomorphism and consider the following assertions: (i) \(\partial([\tilde{f}])=0\in\pi_{m-1}(S^{n-1})\), (ii) \((f,f)\) is loose by small deformation, (iii) \((f,f)\) is loose, (iv) \(N^\#(f,f)=0\), (v) \(E\circ\partial([\tilde{f}])=0\). Then we have that (i) \(\Leftrightarrow\) (ii), (ii) \(\Rightarrow\) (iii) \(\Rightarrow\) (iv), (iv) \(\rightarrow\) (v). Moreover, all of these assertions are equivalent for all maps \(f:M\to N\) if and only if the ``Wecken condition''\ \(0=\partial(\pi_m(S^n)\cap\ker E\) holds. In this case one has that \(\text{MC}(f,f)=\text{MCC}(f,f)=N^\#(f,f)\) for all \(f\). There is a wealth of additional results dealing in particular with \(G=\mathbb{Z}_2\) and the Kervaire invariant.