Abstract
We define an order relation among oriented \(\textit{PD}_4\)-complexes. We show that with respect to this relation, two \(\textit{PD}_4\)-complexes over the same complex are homotopy equivalent if and only if there is an isometry between the second homology groups. We also consider minimal objects of this relation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
We thank the referee for providing us with this argument.
References
Baues, H.J.: Obstruction theory. In: Lecture Notes in Mathematics, vol. 628. Springer, Berlin (1977)
Baues, H.J., Bleile, B.: Poincaré duality complexes in dimension four. Algebraic Geom. Topol. 8, 2355–2389 (2008)
Cavicchioli, A., Hegenbarth, F., Repovš, D.: On minimal Poincaré \(4\) -complexes. Turkish J. Math. 38(3), 535–557 (2014)
Hegenbarth, F., Repovš, D., Spaggiari, F.: Connected sums of \(4\)-manifolds. Topol. Appl. 146(147), 209–225 (2005)
Hillman, J.A.: The algebraic characterization of geometric \(4\)-manifolds. In: London Mathematical Society Lecture Note Series, vol. 198. Cambridge University Press, Cambridge (1994)
Hillman, J.A.: \({\mathit{PD}}_4\)-complexes with strongly minimal models. Topol. Appl. 153, 2413–2424 (2006)
Hillman, J.A.: Strongly minimal \({\mathit{PD}}_4\)-complexes. Topol. Appl. 156, 1565–1577 (2009)
Hillman, J.A.: \({\mathit{PD}}_4\)-complexes and \(2\)-dimensional duality groups. arXiv:1303.5486v4
Robinson, C.A.: Moore-Postnikov systems for non-simple fibrations. Ill. J. Math. 16, 234–242 (1972)
Wall, C.T.C.: Poincaré complexes: I. Ann. Math. 86(2), 213–245 (1967)
Wall, C.T.C.: Surgery on compact manifolds, 2nd edn. American Mathematical Society, Providence (1999) (Edited and with a foreword by Ranicki, A.A.)
Whitehead, J.H.C.: A certain exact sequence. Ann. Math. 52, 51–110 (1950)
Acknowledgments
The authors thank the referees for the clarifications of essential points of the paper and for several suggestions which led to a simplification of the proof of Theorem 5.1. This research was supported by the Slovenian-Turkish grants BI-TR/12-14-001 and 111T667, and Slovenian Research grants P1-0292-0101, J1-5435-0101, and J1-6721-0101.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Cap.
Rights and permissions
About this article
Cite this article
Hegenbarth, F., Pamuk, M. & Repovš, D. Homotopy classification of \(\textit{PD}_4\)-complexes relative an order relation. Monatsh Math 177, 275–293 (2015). https://doi.org/10.1007/s00605-014-0716-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-014-0716-1