Riemannian embeddings in codimension one as unbounded $KK$-cycles

Abstract

Given a codimension one Riemannian embedding of Riemannian spin${}^c$-manifolds $ı:X\to Y$ we construct a family $\{{ı}_{!}^{\epsilon }{\}}_{0<\epsilon <{\epsilon }_0}$ of unbounded $KK$-cycles from $C(X)$ to $C_0(Y)$, each equipped with a connection ${\nabla }^{\epsilon }$ and each representing the shriek class ${ı}_{!}\in KK(C(X),C_0(Y))$. We compute the unbounded product of ${ı}_{!}^{\epsilon }$ 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 $\epsilon$. In the limit $\epsilon \to 0$ the product operator admits an asymptotic expansion of the form $\frac{1}{\epsilon }T+D_X+\mathcal{O}(\epsilon )$ where the ``divergent'' part $T$ is an index cycle representing the unit in $KK(\mathbb{C},\mathbb{C})$ and the constant ``renormalized'' term is the Dirac operator $D_X$ on $X$. The curvature of $({ı}_{!}^{\epsilon },{\nabla }^{\epsilon })$ is further shown to converge to the square of the mean curvature of $ı$ as $\epsilon \to 0$.

15 pages, 1 figure