Noncharacteristic embeddings of the -dimensional torus in the -dimensional torus

Author:
David Miller

Journal:
Trans. Amer. Math. Soc. **342** (1994), 215-240

MSC:
Primary 57Q60; Secondary 57Q35, 57Q45

MathSciNet review:
1179398

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We construct certain exotic embeddings of the *n*-torus in in the standard homotopy class. We turn an embedding characteristic if there exists some map in the standard homotopy class with the property that is the standard coordinate inclusion and . We find examples of noncharacteristic embeddings, *f*, in dimensions , , 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, . Then we can associate to *f* a real-valued signature function on the set of *j*-dimensional unitary representations of , where denotes the fundamental group of the localization of with respect to homology with local coefficients in . 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.

**[APS]**M. F. Atiyah, V. K. Patodi, and I. M. Singer,*Spectral asymmetry and Riemannian geometry. II*, Math. Proc. Cambridge Philos. Soc.**78**(1975), no. 3, 405–432. MR**0397798****[B1]**A. K. Bousfield,*Homological localization towers for groups and Π-modules*, Mem. Amer. Math. Soc.**10**(1977), no. 186, vii+68. MR**0447375****[B2]**A. K. Bousfield,*The localization of spaces with respect to homology*, Topology**14**(1975), 133–150. MR**0380779****[CS1]**Sylvain E. Cappell and Julius L. Shaneson,*The codimension two placement problem and homology equivalent manifolds*, Ann. of Math. (2)**99**(1974), 277–348. MR**0339216****[CS2]**Sylvain E. Cappell and Julius L. Shaneson,*Link cobordism*, Comment. Math. Helv.**55**(1980), no. 1, 20–49. MR**569244**, 10.1007/BF02566673**[CS3]**Sylvain E. Cappell and Julius L. Shaneson,*An introduction to embeddings, immersions and singularities in codimension two*, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 129–149. MR**520529****[CO1]**T. D. Cochran and K. E. Orr,*Not all links are concordant to boundary links*, Bull. Amer. Math. Soc. (N.S.)**23**(1990), no. 1, 99–106. MR**1031581**, 10.1090/S0273-0979-1990-15912-9**[CO2]**-,*Not all links are concordant to boundary links*, preprint.**[HS]**Wu-chung Hsiang and Julius L. Shaneson,*Fake tori*, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 18–51. MR**0281211****[LD]**J. Le Dimet,*Cobordisme d'enlacement de disques*, Mem. Math. France**116**(1988).**[L1]**J. Levine,*Finitely-presented groups with long lower central series*, Israel J. Math.**73**(1991), no. 1, 57–64. MR**1119927**, 10.1007/BF02773424**[L2]**J. P. Levine,*Link concordance and algebraic closure. II*, Invent. Math.**96**(1989), no. 3, 571–592. MR**996555**, 10.1007/BF01393697**[L3]**-,*Signature invariants of homology bordism with applications to links*, Proc. Osaka Knot Theory Conf.**[L4]**J. P. Levine,*Link concordance and algebraic closure of groups*, Comment. Math. Helv.**64**(1989), no. 2, 236–255. MR**997364**, 10.1007/BF02564673**[L5]**J. P. Levine,*Algebraic closure of groups*, Combinatorial group theory (College Park, MD, 1988) Contemp. Math., vol. 109, Amer. Math. Soc., Providence, RI, 1990, pp. 99–105. MR**1076380**, 10.1090/conm/109/10**[L6]**-,*Link invariants via the eta invariant*, preprint.**[V]**Pierre Vogel,*Localization of spaces with respect to a class of maps*, preprint.**[W]**C. T. C. Wall,*Surgery on compact manifolds*, Academic Press, London-New York, 1970. London Mathematical Society Monographs, No. 1. MR**0431216**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57Q60,
57Q35,
57Q45

Retrieve articles in all journals with MSC: 57Q60, 57Q35, 57Q45

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1179398-7

Article copyright:
© Copyright 1994
American Mathematical Society