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

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- by David Miller PDF
- Trans. Amer. Math. Soc.
**342**(1994), 215-240 Request permission

## 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.

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## Additional Information

- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**342**(1994), 215-240 - MSC: Primary 57Q60; Secondary 57Q35, 57Q45
- DOI: https://doi.org/10.1090/S0002-9947-1994-1179398-7
- MathSciNet review: 1179398