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Higher \(\rho\)-invariants. (English) Zbl 0946.57037

Farber, Michael (ed.) et al., Tel Aviv topology conference: Rothenberg Festschrift. Proceedings of the international conference on topology, Tel Aviv, Israel, June 1-5, 1998 dedicated to Mel Rothenberg on the occasion of his 65th birthday. Providence, RI: American Mathematical Society. Contemp. Math. 231, 315-320 (1999).
An \(n\)-dimensional manifold \(M\) is said to be antisimple if the chain complex \(C_{\ast}(M)\) of \(M\) is chain equivalent to a complex of projective modules \(P_{\ast}\) with \(P_m = 0\) for \(m\) in the middle dimension (\(n=2m\) or \(2m+1\)). Given an antisimple \(n\)-dimensional manifold \(M\) with the fundamental group \(\pi_1(M) = \Gamma\) for which the Novikov conjecture is true, the author associates to \(M\) an invariant \[ H\rho(M) \in L_{n+1}(\Gamma)\otimes {\mathbb Q}/A_{\Gamma} \biggl(\bigoplus_{i\geq 0} H_{n+1-4i}(B\Gamma;{\mathbb Q})\biggr) \] called \(\rho\)-invariant of \(M\), where \(A_{\Gamma}: \bigoplus_{i\geq 0} H_{\ast+1-4i}(B\Gamma;{\mathbb Q}) \rightarrow L_{\ast}(\Gamma) \otimes {\mathbb Q}\) is the rational Quinn-Ranicki assembly map for the classifying space \(B\Gamma\). The Weinberger \(\rho\)-invariant ranges over a lattice in the rational cobordism group of \((n+1)\)-dimensional Poincaré complexes (in fact, the \({\mathbb Q}\)-vector space) \(L_{n+1}(\Gamma) \otimes {\mathbb Q} = L^{n+1}(\Gamma) \otimes {\mathbb Q}\) modulo \(im(A_{\Gamma})\) as one varies among manifolds homotopy equivalent to \(M\). This invariant is an analogue of the original Atiyah-Bott-Singer \(\rho\)-invariant of odd-dimensional manifolds with finite fundamental group, which occurs in classifying lens spaces and their homotopy analogues.
Roughly, \(H\rho(M) = k^{-1}(kP^{<m} \cup IC_{\ast}^k(W)) \in L^{n+1}(\Gamma)\) is an \((n+1)\)-dimensional algebraic Poincaré complex. It is obtained by glueing together in the sense of Ranicki two algebraic Poincaré complexes. The first one is just \(P_{\ast}\) truncated at \(m\) because \(M\) is antisimple. The second one is more geometric, and follows the author’s trick from [Proc. Natl. Acad. Sci. USA 85, No. 15, 5362-5363 (1988; Zbl 0659.57016)] applied to study the ordinary \(\eta\)-invariant. The author uses the bordism group \(\Omega_{n}^{\text{Witt}}\) of Witt spaces in the sense of P. H. Siegel [Am. J. Math. 105, 1067-1105 (1983; Zbl 0547.57019)]. Since the Novikov conjecture is true for \(\Gamma\), the rational injectivity of the assembly map \(A_{\Gamma}\) and the fact that \(\Omega_{n}^{\text{Witt}}(B\Gamma) \otimes {\mathbb Q} \simeq \bigoplus_{i\geq 0} H_{n-4i}(B\Gamma;{\mathbb Q})\) implies that the Witt map \(\Omega_{n}^{\text{Witt}}(B\Gamma) \rightarrow L^n(\Gamma) \otimes {\mathbb Q}\), defined using symmetric signature for Witt spaces by S. Cappell, J. Shaneson and the author [C. R. Acad. Sci., Paris, Sér. I 313, No. 5, 293-295 (1991; Zbl 0742.57023)], is rationally injective, and that therefore, some multiple \(kM\) of \(M\) bounds a Witt space \(W\) such that \(\pi_1(M) \rightarrow \pi_1(W)\) is an isomorphism. In the construction, \(IC_{\ast}(W)\) is the complex of intersection chains of \(W\) in the sense of Goresky-MacPherson.
For the entire collection see [Zbl 0913.00045].

MSC:

57R67 Surgery obstructions, Wall groups
19J25 Surgery obstructions (\(K\)-theoretic aspects)
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
57N70 Cobordism and concordance in topological manifolds
57P10 Poincaré duality spaces
57Q20 Cobordism in PL-topology
55N33 Intersection homology and cohomology in algebraic topology
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
55U15 Chain complexes in algebraic topology
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