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We study the index theorem and Chern character of an admissible pseudomanifold X† with one singular stratum. Under a condition on the link, we give a de Rham type realization of the Goresky-MacPherson-Siegel L-classes on X† in terms of curvature forms and eta invariant of the link. 0. Introduction In [CST] and [MW1] Connes-Sullivan-Teleman and Moscovici-Wu solved a longstanding problem of recovering the topological Pontryagin classes from local data. In [MW1] and [MW2], using the finite propagation speed property in the unbounded picture, Moscovici-Wu gave a realization of the L-classes, hence of the Pontryagin classes, of the topological manifold in terms of Alexander-Spanier cycles by looking at the “straight” Chern character of [CM]. In this paper, we will generalize the result in [MW1] to a particular type of pseudomanifolds, namely X† = M ∪ (c†(L)×N). As in [C1] and [GM], for spaces with singularities, we work with “characteristic classes” in homology rather than cohomology. In [GM] Goresky and MacPherson defined L-classes for pseudomanifolds with even codimensional strata. By the work of Siegel [Si], one can extend this definition to Witt spaces. Our approach differs from [MW1] in the sense that we relate the “straight” Chern character of D to L-class directly. This asserts that (Theorem 3.4) the Goresky-MacPherson-Siegel L-class of X† (with 2L having zero oriented cobordism) is represented by the following cycle: L∗(X†; ρX†) = 2 2mLm′(R(gM ))⊕−δ` η(L) 22nLn′(R(gN )), where m′ = m−∗ 4 , n ′ = n−∗ 4 and δ` = 1 2 (1− (−1)`). Furthermore, we will show that the Chern character of D coincides with the “straight” Chern character of D. In [BC] and [C2], Bismut and Cheeger defined L-classes by means of the index pairing. Theorem 4.1 below shows that the Bismut-Cheeger L-classes and GoreskyMacPherson-Siegel L-classes are the same (up to constants) for spaces X† defined above. Received by the editors January 17, 1995 and, in revised form, March 13, 1995 and January 31, 1996. 1991 Mathematics Subject Classification. Primary 19D55; Secondary 58G12. c ©1997 American Mathematical Society

$\mathcal {L}$-classes on pseudomanifolds with one singular stratum