Higher-order Nielsen numbers (Q2570908)

The author studies the coincidence problem for maps \(f,g:X\to Y\) for manifolds in the case of arbitrary codimension \(\dim X-\dim Y\). More generally, let \(X\), \(Y\), and \(Z\) be connected CW-complexes, \(Y\subset Z\) and \(f:X\to Z\) a map (with the obvious specifications this yields the fixed point and the coincidence point problem as special cases). Let \(C=f^{-1}(Y)\). Denote by \(M_p(C)\) the set of all singular orientable compact closed \(p\)-manifolds, i.e., maps \(s:S\to C\) where \(S\) is an orientable closed compact \(p\)-manifold. Two singular \(p\)-manifolds \(s_0:S_0\to C\) and \(s_1:S_1\to C\) are said to be Nielsen equivalent if (1) \(js_0\), \(js_1\) (\(j:C\to X\) is the inclusion) are bordant in \(X\) via a map \(H:W\to X\) extending \(s_0\sqcup s_1\) such that \(W\) is a bordism between \(S_0\) and \(S_1\), (2) \(fs_0\) and \(fs_1\) are bordant in \(Y\) via a map \(G:W\to Y\) extending \(fs_0\sqcup fs_1\), (3) \(fH\) and \(G\) are homotopic relative to \(S_0\sqcup S_1\). Denote the Nielsen class of \(s\in M_p(C)\) by \([s]_N\). The corresponding quotient group \(S'_p(f,Y)\) is called the group of Nielsen classes of order \(p\). Moreover, Nielsen equivalence is an equivalence in the group \(\Omega_*(C)\) of bordism classes. Let \(S'_*(f,Y)\) denote the corresponding quotient group of \(\Omega_*(C)\). If \(F:I\times X\to Z\) is a homotopy, \(i_t:X\to\{t\}\times X\to I\times X\) the obvious inclusion, \(f_t:=Fi_t\), then \({i'_t}_*:S'_*(f_t)\to S'_*(F)\) is a monomorphism, so \(M_*^F=\text{Im}{i'_0}_*\cap\text{Im}{i'_1}_*\) can be viewed as a subgroup of \(S'_*(f_0)\) and one defines the group of essential Nielsen classes \(S_*(f,Y)\) to be the intersection of all \(M_*^F\) where \(F\) is a homotopy of \(f\). The Nielsen number of \(N_p(f,Y)\) is defined as the rank of \(S_p(f,Y)\). If \(f\in\Omega_q(C)\) one defines the index of \([s]_N\in S'_q(f)\) as \(I_f(s):=f_*(s)\) where \(f_*:\Omega_q(C)\to\Omega_q(Y)\). \(I_f:S'_*(f)\to\Omega_*(Y)\) is a homomorphism, so one defines the group of algebraically essential Nielsen classes as \(S_*^a(f,Y):=S'_*(f,Y)/\ker I_f\). It turns out that every algebraically essential class \(z\) is topological essential, i.e., \(z\) cannot be reduced by a homotopy to the zero class. The author discusses earlier approaches to higher order Nielsen theory and he discusses several examples. There is also a section on Wecken numbers in the codimension 1 case.