We construct new commutative diagrams of exact sequences which relate surgery and splitting obstruction groups for pairs of manifolds. The splitting and surgery obstruction groups are computed for pairs of manifolds and various geometric diagrams of groups corresponding to the problem of splitting along a one-sided submanifold of codimension 1.
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P. M. Akhmet’ev, “Splitting of homotopic equivalences along a one-sided submanifold of codimension 1,” Izv. Akad. Nauk SSSR, Ser. Mat., 51, 211–241 (1987).
P. M. Akhmet’ev and Yu. V. Muranov, “Splitting obstruction for manifolds with infinite fundamental group,” Mat. Zametki, 60, Issue 2, 163–175 (1996).
A. Bak and Yu. V. Muranov, “Splitting along submanifolds, and L-spectra,” Sovrem. Mat. Prilozh. Topol., Anal. Smezh. Vopr., No. 1, 3–18 (2003).
A. Bak and Yu. V. Muranov, “Normal invariants of manifold pairs and assembly maps,” Mat. Sb., 197, No. 6, 3–24 (2006).
W. Browder and G. R. Livesay, “Fixed point free involutions on homotopy spheres,” Bull. Amer. Math. Soc., 73, 242–245 (1967).
S. E. Cappell and J. L. Shaneson, “Pseudo-free actions. I,” in: Lecture Notes in Mathematics, 763 (1979), pp. 395–447.
A. Cavicchioli, Yu. V. Muranov, and F. Spaggiari, “Relative groups in surgery theory,” Bull. Belg. Math. Soc. Simon Stevin., 12, No. 12, 109–113 (2005).
A. Cavicchioli, Yu. V. Muranov, and F. Spaggiari, “Surgery on pairs of closed manifolds,” Czechoslovak Math. J., 59, No. 134, 551–571 (2009).
M. Cencelj, Yu. V. Muranov, and D. Repovs, ”On structure sets of manifold pairs,” Homology, Homotopy Appl., 11, 195–222 (2009).
I. Hambleton, A. Ranicki, and L. Taylor, “Round L-theory,” J. Pure Appl. Algebra, 47, 131–154 (1987).
F. Hegenbarth, Yu. V. Muranov, and D. Repovš, “Browder–Livesay invariants and the example of Cappell and Shaneson,” Milan J. Math., 80 (2012).
Yu. V. Muranov, “Obstruction groups to splitting and quadratic extensions of antistructures,” Izv. Ros. Akad. Nauk, Ser. Mat., 59, 107–132 (1995).
Yu. V. Muranov and A. F. Kharshiladze, “Browder–Livesay groups of Abelian 2-groups,” Mat. Sb., 181, No. 8, 1061–1098 (1990).
Yu. V. Muranov and D. Repovš, “Obstruction groups for surgeries and splitting for a pair of manifolds,” Mat. Sb., 188, No. 3, 127–142 (1997).
Yu. V. Muranov and D. Repovš, “LS-groups and morphisms of quadratic extensions,” Mat. Zametki, 70, 419–424 (2001).
A. A. Ranicki, “The total surgery obstruction,” in: Lecture Notes in Mathematics, 763 (1979), pp. 275–316.
A. A. Ranicki, Exact Sequences in the Algebraic Theory of Surgery, Princeton University Press, Princeton (1981).
B. Ruini and F. Spaggiari, “On the computation of L-groups and natural maps,” Abh. Math. Semin. Univ. Hamburg, 72, 297–308 (2002).
C. T. C. Wall, “Classification of Hermitian forms. VI. Group rings,” Ann. Math., 103, 1–80 (1976).
C. T. C. Wall, Surgery on Compact Manifolds, American Mathematical Society, Providence (1999).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 3, pp. 316–332, March, 2014.
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Muranov, Y.V., Jiménez, R. Splitting Obstruction Groups Along one-Sided Submanifolds. Ukr Math J 66, 352–370 (2014). https://doi.org/10.1007/s11253-014-0936-5
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DOI: https://doi.org/10.1007/s11253-014-0936-5