Noncharacteristic embeddings of the $n$-dimensional torus in the $(n+2)$-dimensional torus

Abstract:

We construct certain exotic embeddings of the n-torus ${T^n}$ in ${T^{n + 2}}$ in the standard homotopy class. We turn an embedding $f:{T^n} \to {T^{n + 2}}$ characteristic if there exists some map $\alpha :{T^{n + 2}} \to {T^{n + 2}}$ in the standard homotopy class with the property that $\alpha \; \circ \;f:{T^n} \to {T^{n + 2}}$ is the standard coordinate inclusion and $\alpha ({T^{n + 2}} - f({T^n})) \subset {T^{n + 2}} - {T^n}$. We find examples of noncharacteristic embeddings, f, in dimensions $n = 4k + 1$, $n \geq 5$, and show that these examples are not even cobordant to characteristic embeddings. We let G denote the fundamental group of the complement of the standard coordinate inclusion, ${T^{n + 2}} - {T^n}$. Then we can associate to f a real-valued signature function on the set of j-dimensional unitary representations of $\bar G$, where $\bar G$ denotes the fundamental group of the localization of ${T^{n + 2}} - {T^n}$ with respect to homology with local coefficients in $\mathbb {Z}[{\mathbb {Z}^{n + 2}}]$. This function is a cobordism invariant which has certain periodicity properties for characteristic embeddings. We verify that this periodicity does not hold for our examples, f, implying that they are not characteristic. Additional results include a proof that the examples, f, become cobordant to characteristic embeddings upon taking the cartesian product with the identity map on a circle.