Abstract
Let M 3 be a 3-dimensional manifold with fundamental group π 1(M) which contains a quaternion subgroup Q of order 8. In 1979 Cappell and Shaneson constructed a nontrivial normal map f : M 3 × T 2 → M 3 × S 2 which cannot be detected by simply connected surgery obstructions along submanifolds of codimension 0, 1, or 2, but it can be detected by the codimension 3 Kervaire-Arf invariant. The proof of non-triviality of \({\sigma (f) \in L_{5}(\pi _{1}(M))}\) is based on consideration of a Browder-Livesay filtration of a manifold X with \({\pi _{1}(X) \cong \pi _{1}(M)}\) . For a Browder-Livesay pair \({Y^{n-1} \subset X^{n}}\), the restriction of a normal map to the submanifold Y is given by a partial multivalued map Γ : L n (π 1(X)) → L n−1(π 1(Y)), and the Browder-Livesay filtration provides an iteration Γn. This map is a basic step in the definition of the iterated Browder-Livesay invariants which give obstructions to realization of surgery obstructions by normal maps of closed manifolds. In the present paper we prove that Γ 3(σ (f)) = 0 for any Browder-Livesay filtration of a manifold X 4k+1 with \({\pi _{1}(X) \cong Q}\) . We compute splitting obstruction groups for various inclusions ρ → Q of index 2, describe natural maps in the braids of exact sequences, and make more precise several results about surgery obstruction groups of the group Q.
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Cappell S.E., Shaneson J.L.: A counterexample on the oozing problem for closed manifolds. Lecture Notes in Math. 763, 627–634 (1979)
Cappell S.E., Shaneson J.L.: Pseudo-free actions. I. Lecture Notes in Math. 763, 395–447 (1979)
Hambleton I.: Projective surgery obstructions on closed manifolds. Lecture Notes in Math. 967, 101–131 (1982)
I. Hambleton, A.F. Kharshiladze, A spectral sequence in surgery theory. Mat. Sbornik 183 (1992), 3–14; English transl. in Russian Acad. Sci. Sb. Math. 77 (1994), 1–9.
A.F. Kharshiladze, Iterated Browder-Livesay invariants and oozing problem. Mat. Zametki 41 (1987), 557–563; English transl. in Math. Notes 41 (1987), 312–315.
Yu.V. Muranov, Relative Wall groups and decorations. Mat. Sbornik 185 (1994), 79–100; English. transl. in Russian Acad. Sci. Sb. Math. 83 (1995), 495–514.
Yu.V. Muranov, D. Repovš, R. Jimenez, Surgery spectral sequence and manifolds with filtration. Trudy MMO 67 (2006), 294–325; English translation in Trans. Moscow Math. Soc. 67 (2006), 261–288.
Ranicki A.A.: The L-theory of twisted quadratic extensions. Canadian J. Math. 39, 345–364 (1987)
C.T.C. Wall, Surgery on Compact Manifolds. Second Edition, Amer. Math. Soc., Providence, R.I., 1999.
Wall C.T.C.: Classification of Hermitian forms. VI Group rings. Ann of Math. 103, 1–80 (1976)
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Hegenbarth, F., Muranov, Y.V. & Repovš, D. Browder-Livesay Filtrations and the Example of Cappell and Shaneson. Milan J. Math. 81, 79–97 (2013). https://doi.org/10.1007/s00032-012-0192-9
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DOI: https://doi.org/10.1007/s00032-012-0192-9