The Borel regulator map on pictures. II: An example from Morse theory.

*(English)*Zbl 0793.19002Suppose that there is a smooth bundle \(E\) over an even dimensional sphere \(S^{2k}\) and a unitary representation of \(\pi_ 1 E\) with respect to which each fiber becomes acyclic. Then a fiberwise “framed” function on \(E\) gives a family of acyclic chain complexes over \(\mathbb{C}\) parametrized by \(S^{2k}\). By one version of algebraic \(K\)-theory this gives an element of \(K_{2k+1}\mathbb{C}\) modulo some indeterminacy. However by applying the Borel regulator map \(b_ k : K_{2k+1} \mathbb{C}\to \mathbb{R}\) one obtains a well defined real valued invariant of the bundle \(E\to S^{2k}\) which is called its higher Reidemeister torsion since it agrees with the usual Reidemeister torsion of \(E\) (actually difference between the torsion of the two components) in the case when \(k=0\).

This paper gives a detailed analysis of the simplest example of this, namely, a circle bundle \(S^ 1\to E\to S^ 2\). The authors choose a specific fiberwise generalized Morse function on \(E\) and obtain a family of invertible matrices with coefficients in \(\mathbb{Z}[\pi_ 1 E]\). (The determinant of each matrix is \(1-u\) where \(u\) is the generator of \(\pi_ 1 E= \mathbb{Z}/n\).) These matrices vary by elementary row and column operations and the authors go through some lengthy technicalities in order to reduce these to column operations only.

A two parameter family of invertible matrices changing by column operations gives what is called a “picture” and represents an element of \(K_ 3\) of the ring of coefficients. In this case the authors obtain an element \(L_ n\) of \(K_ 3 \mathbb{Z}[\xi]\) where \(\xi\) is any nontrivial \(n\)-th root of unity. Using an explicit dilogarithm formula for the Borel regulator on pictures derived in the first part of the paper (see the review above), they are able to compute the higher Reidemeister torsion invariant and they obtain \(b(L_ n)=n \text{ Im dilog}(\xi)\).

Bismut and Lott have given an analytic definition of higher Reidemeister torsion and computed it in many cases. For circle bundles over \(S^ 2\) their calculation also gives \(n\text{ Im dilog}(\xi)\). It is not known whether the two definitions of higher Reidemeister torsion agree in other cases.

This paper gives a detailed analysis of the simplest example of this, namely, a circle bundle \(S^ 1\to E\to S^ 2\). The authors choose a specific fiberwise generalized Morse function on \(E\) and obtain a family of invertible matrices with coefficients in \(\mathbb{Z}[\pi_ 1 E]\). (The determinant of each matrix is \(1-u\) where \(u\) is the generator of \(\pi_ 1 E= \mathbb{Z}/n\).) These matrices vary by elementary row and column operations and the authors go through some lengthy technicalities in order to reduce these to column operations only.

A two parameter family of invertible matrices changing by column operations gives what is called a “picture” and represents an element of \(K_ 3\) of the ring of coefficients. In this case the authors obtain an element \(L_ n\) of \(K_ 3 \mathbb{Z}[\xi]\) where \(\xi\) is any nontrivial \(n\)-th root of unity. Using an explicit dilogarithm formula for the Borel regulator on pictures derived in the first part of the paper (see the review above), they are able to compute the higher Reidemeister torsion invariant and they obtain \(b(L_ n)=n \text{ Im dilog}(\xi)\).

Bismut and Lott have given an analytic definition of higher Reidemeister torsion and computed it in many cases. For circle bundles over \(S^ 2\) their calculation also gives \(n\text{ Im dilog}(\xi)\). It is not known whether the two definitions of higher Reidemeister torsion agree in other cases.

Reviewer: K.Igusa (Waltham)

##### MSC:

19D55 | \(K\)-theory and homology; cyclic homology and cohomology |

58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |

19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |

##### Keywords:

Morse theory; pseudoisotopy; picture; algebraic \(K\)-theory; Borel regulator map; Reidemeister torsion; dilogarithm
PDF
BibTeX
XML
Cite

\textit{K. Igusa} and \textit{J. Klein}, \(K\)-Theory 7, No. 3, 225--267 (1993; Zbl 0793.19002)

Full Text:
DOI

##### References:

[1] | Bloch, S.: Higher regulators, algebraicK-theory, and zeta functions of elliptic curves, Irvine Lecture Notes (1978). |

[2] | Beilinson, A. A.: Higher regulators and values ofL-functions,Sovrem. Probl. Mat. 24, Viniti, Moscow (1984). Translation inJ. Soviet. Math. 30 (1985), 2036-2070. · Zbl 0588.14013 |

[3] | Cerf, J.: La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie,I.H.E.S. Publ. Math. 39 (1970), 5-173. · Zbl 0213.25202 |

[4] | Esnault, H.: On the Loday Symbol in Deligne-Beilinson Cohomology,K-Theory 3 (1989), 1-28. · Zbl 0697.14006 |

[5] | Hsiang, W-C.: On ? i (Diff(M n )), in J. Cantrell (ed),Geometric Topology, Academic Press, New York, 1979, pp. 351-365. · Zbl 0493.57014 |

[6] | Hatcher, A. and Wagoner, J.: Pseudo-isotopies of compact manifolds,Astérisque 6 (1973). · Zbl 0274.57010 |

[7] | Igusa, K.: The Wh3(?) obstruction for pseudoisotopy, PhD Thesis, Princeton University, 1979. |

[8] | Igusa, K.: On the homotopy type of the space of generalized Morse functions,Topology 23 (1984), 245-256. · Zbl 0595.57025 |

[9] | Igusa, K.: The stability theorem for smooth pseudoisotopies,K-Theory 2 (1988), 1-355. · Zbl 0691.57011 |

[10] | Igusa, K. and Klein, J.: Filtered chain complexes and higher Franz-Reidemeister torsion, in preparation. |

[11] | Klein, J.: The cell complex construction and higherR-torsion for bundles with framed Morse function, PhD. Thesis, Brandeis University, 1989. |

[12] | Klein, J.: Parametrized Morse theory and higher Franz-Reidemeister torsion, preprint, Universität GH-Siegen (1991). |

[13] | Lee, R. and Szczarba, R.: The groupK 3(?) is cyclic of order forty-eight,Ann. Math. (2)104 (1976), 31-60. · Zbl 0341.18008 |

[14] | Neukirch, J.: The Beilinson conjecture for algebraic number fields, inBeilinson’s Conjectures on Special Values of L-Functions, Perspectives in Math. 4, Academic Press, New York, 1988, pp. 193-247. |

[15] | Ramakrishnan, D.: Regulators, algebraic cycles, values ofL-functions,Algebraic K-Theory and Algebraic Number Theory, Contemp. Math. Vol. 83, 1988, pp. 183-310. |

[16] | Soulé, C.: Eléments cyclotomique enK-théorie,Astérisque 147-148 (1987), 225-257. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.