Let \(X\) be a metrizable compact singular space with open dense submanifold \(M\), and let \(Y=X\setminus Y\). The short exact sequence of \(C^*\) algebras \[ 0\to C_0(M)\to C(X)\to C(Y)\to0 \] induces a six-term exact sequence in \(K\)-homology, which contains the following pieces: \[ \cdots\to KK_i(C(X),\mathbb{C})\to KK_i(C_0(M),\mathbb{C})\to KK_{i-1}(C(Y),\mathbb{C})\to\cdots \] for \(i\in\{0,1\}\) (the last map is the connecting homomorphism denoted by \(\partial\)). The notation \(K_i(X)\) and \(K_i(Y)\) can be used for \(KK_i(C(X),\mathbb{C})\) and \(KK_i(C(Y),\mathbb{C})\) because there is only one notion of \(K\)-homology for a compact metrizable space. Assume that \(M\) is equipped with a complete Riemannian metric and a spin structure, and let \(D\) be the corresponding Dirac operator acting on sections of the spinor bundle \(S\) over \(M\). Then \((L^2(S),D(1+D^*D)^{-1/2})\) represents a class \([D]_M\in KK_*(C_0(M),\mathbb{C})\). The first goal of the paper is to give conditions so that \([D]_M\) represents a class in \(K_*(X)\), which is equivalent to \(\partial([D]_M)=0\) in \(K_*(Y)\) according to the above exact sequence in \(K\)-homology. For instance, it is proved that this holds when the scalar curvature of \(M\) is properly positive and \(C(X)\) contains a dense subset of functions whose restrictions to \(M\) are smooth with bounded exterior derivative. Conditions are also given to get the same class in \(K_*(X)\) by using two different complete metrics on \(M\).
This work can be seen as part of a general program to get topological (or geometrical) invariants of \(X\) by using \(L^2\) analysis on \(M\).