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Intersection R-torsion and analytic torsion for pseudomanifolds. (English) Zbl 0605.57012

For a closed Riemannian manifold the Reidemeister torsion equals the analytic torsion T [see J. Cheeger, Ann. Math., II. Ser. 109, 259- 322 (1979; Zbl 0412.58026); W. Müller, Adv. Math. 28, 233-305 (1978; Zbl 0395.57011)]. We construct a finer invariant, the intersection R-torsion \(I\tau\) for pseudomanifolds. For metrics with conical singularities we study the analytic torsion T. Our study indicates that in this case also the intersection R-torsion \(I\tau\) equals the analytic torsion T.

MSC:

57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57P05 Local properties of generalized manifolds
58J99 Partial differential equations on manifolds; differential operators
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References:

[1] Chapman, T.A.: Compact Hilbert cube manifolds and the invariance of Whitehead torsion. Bull. Am. Math. Soc., New Ser.79, 52-56 (1973) · Zbl 0251.57004 · doi:10.1090/S0002-9904-1973-13087-3
[2] Chapman, T.A.: Topological invariance of Whitehead torsion. Am. J. Math.96, 488-497 (1974) · Zbl 0358.57004 · doi:10.2307/2373556
[3] Cheeger, J.: Analytic torsion and the heat equation. Ann. Math.109, 259-322 (1979) · Zbl 0412.58026 · doi:10.2307/1971113
[4] Cheeger, J.: On the spectral geometry of spaces with cone-line singularities. Proc. Natl. Acad. Sci. USA76, 2103-2106 (1979) · Zbl 0411.58003 · doi:10.1073/pnas.76.5.2103
[5] Cheeger, J.: On the Hodge theory of Riemannian pseudomanifolds. Proc. Symp. Pure Math.36, 91-145 (1980) · Zbl 0461.58002
[6] Cheeger, J.: Spectral geometry of singular Riemannian spaces. J. Differ. Geom.18, 575-657 (1983) · Zbl 0529.58034
[7] Cheeger, J., Goresky, M., Macpherson, R.: Intersection homology andL 2-cohomology of algebraic varieties. Ann. Math. Studies No. 102, Princeton, NJ: Princeton University Press 1982 · Zbl 0503.14008
[8] Gilkey, P.: The index theorem and the heat equation. Publish or Perish, 1974 · Zbl 0287.58006
[9] Goresky, M., Macpherson, R.: Intersection homology theory. Topology19, 135-162 (1980) · Zbl 0448.55004 · doi:10.1016/0040-9383(80)90003-8
[10] Goresky, M., Macpherson, R.: Intersection homology II. Invent. Math.72, 77-130 (1983) · Zbl 0529.55007 · doi:10.1007/BF01389130
[11] Kwun, K.W., Szczarba, R.H.: Product and sum theorems for Whitehead torsion. Ann. Math.82, 183-190 (1965) · Zbl 0142.40703 · doi:10.2307/1970568
[12] Milnor, J.: Whitehead torsion. Ball A.M.S.72, 348-426 (1966)
[13] Milnor, J.: A duality theorem for Whitehead torsion. Ann. Math.76, 137-147 (1962) · Zbl 0108.36502 · doi:10.2307/1970268
[14] Müller, W.: Analytic torsion andR-torsion of Riemannian manifolds. Adv. Math.28, 233-305 (1978) · Zbl 0395.57011 · doi:10.1016/0001-8708(78)90116-0
[15] Ray, D., Singer, I.:R-torsion and the Laplacian on Riemannian manifolds. Adv. Math.71, 145-210 (1971) · Zbl 0239.58014 · doi:10.1016/0001-8708(71)90045-4
[16] Rourke, C.P., Sanderson, B.T.: Introduction to Piecewise-Linear Topology. Berlin, Heidelberg, New York: Springer 1982 · Zbl 0477.57003
[17] Steenrod, N.: The Topology of fibre bundles. Princeton: University Press 1951 · Zbl 0054.07103
[18] Warner, F.W.: Foundations of differentiable manifolds and Lie groups. Scott, Foresman 1971 · Zbl 0241.58001
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