On the Künneth formula for reduced \(L_2\)-cohomologies
It is well known that the Künneth formula \(H^k_2(X\times Y)\cong \bigoplus_{i+ j=k} H^i_2(X)\otimes H^j_2(Y)\) is valid for the \(L_2\)-cohomologies of the product of two Riemannian manifolds in case one of the manifolds \(X\) and \(Y\) is compact. For the warped products \(X\times_f Y\), the following result holds: if the warping function \(f\) is bounded and the operator of the exterior differentiation \(d_Y:L^{j- 1}_2(Y)\to L^j_2(Y)\) is normally solvable for each \(1\leq j\leq n=\dim Y\), i.e., it has closed range, then \[ H^k_2(X\times_f Y)\cong H^i_2(X, f^{n/2- j}; H^j(Y)).\tag{1} \] Formula (1) expressing the \(L_2\)-cohomologies of the warped product \(X\times_f Y\) in terms of the weighted cohomologies of the manifold \(X\) with weights \(f^{n/2-j}\) and coefficients in the Hilbert spaces \(H^j(Y)\) was established by \textit{S. Zucker} [Invent. Math. 70, 169-218 (1982; Zbl 0508.20020)] under some additional assumption on the domain of the operator \(d_{X\times_f Y}\). In [\textit{V. M. Gol'dshtejn} and the authors, Sib. Math. J. 32, No. 5, 749-760 (1991); translation from Sib. Mat. Zh. 32, No. 5(189), 29-42 (1991; Zbl 0754.55017)] formula (1) was proved without constraints in the case of \(L_p\)-cohomologies with \(1< p<\infty\). Isomorphism (1) is a topological isomorphism; therefore, it induces an isomorphism \[ \overline H^k_2(X\times_f Y)\cong \bigoplus_{i+j=k} \overline H^i_2(X, f^{n/2-j};\overline H^j_2(Y))\tag{2} \] of reduced \(L_2\)-cohomologies. The purpose of the present article is to prove formula (2) in the case of a bounded warping function \(f\) without the condition of normal solvability of the operator \(d_Y\). We note that the proof uses the spectral theorem for selfadjoint operators in a Hilbert space and therefore it cannot be transferred to the case of \(L_p\)-cohomologies with \(p\neq 2\).
scientific article; zbMATH DE number 923482