Abstract

Given a codimension one Riemannian embedding of Riemannian spin${}^c$-manifolds $ı:X\to Y$ we construct a family ${\{{ı}_{!}^{𝜖}\}}_{0<𝜖<{𝜖}_0}$ of unbounded $KK$-cycles from $C(X)$ to $C_0(Y)$, each equipped with a connection ${\nabla <!--FUNCTION\; APPLICATION-->}^{𝜖}$ and each representing the shriek class ${ı}_{!}\in KK(C(X),C_0(Y))$. We compute the unbounded product of ${ı}_{!}^{𝜖}$ with the Dirac operator $D_Y$ on $Y$ and show that this represents the $KK$-theoretic factorization of the fundamental class $[X]={ı}_{!}\otimes [Y]$ for all $𝜖$. In the limit $𝜖\to 0$ the product operator admits an asymptotic expansion of the form $\frac{1}{𝜖}T+D_X+(𝜖)$ where the “divergent” part $T$ is an index cycle representing the unit in $KK(ℂ,ℂ)$ and the constant “renormalized” term is the Dirac operator $D_X$ on $X$. The curvature of $({ı}_{!}^{𝜖},{\nabla <!--FUNCTION\; APPLICATION-->}^{𝜖})$ is further shown to converge to the square of the mean curvature of $ı$ as $𝜖\to 0$.

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