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On minimal Poincare 4-complexes

We consider 2 types of minimal Poincare 4 -complexes. One is defined with respect to the degree 1-map order. This idea was already present in our previous papers, and more systematically studied later by Hillman. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2014) 38: 535 – 557 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1211-54 On minimal Poincar´ e 4-complexes 1 ˘ 3,∗ Alberto CAVICCHIOLI1 , Friedrich HEGENBARTH2 , Du˘ san REPOVS Department of Mathematics, University of Modena and Reggio Emilia, Modena, Italy 2 Department of Mathematics, University of Milan, Milan, Italy 3 Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia Received: 28.11.2012 • Accepted: 25.06.2013 • Published Online: 14.03.2014 • Printed: 11.04.2014 Abstract: We consider 2 types of minimal Poincar´e 4 -complexes. One is defined with respect to the degree 1 -map order. This idea was already present in our previous papers, and more systematically studied later by Hillman. The second type of minimal Poincar´e 4 -complexes was introduced by Hambleton, Kreck, and Teichner. It is not based on an order relation. In the present paper we study existence and uniqueness questions. Key words: Poincar´e 4 -complex, equivariant intersection form, degree 1 -map, k -invariant, homotopy type, obstruction theory, homology with local coefficients, Whitehead’s quadratic functor, Whitehead’s exact sequence 1. Introduction Minimal objects are usually defined with respect to a partial order. We consider oriented Poincar´e 4 -complexes (in short, PD4 -complexes). If X and Y are 2 PD4 -complexes, we define X ≻ Y if there is a degree 1 -map f : X → Y inducing an isomorphism on the fundamental groups. If also Y ≻ X , well-known theorems imply that f : X → Y is a homotopy equivalence. So ”≻” defines a symmetric partial order on the set of homotopy types of PD4 -complexes. A PD4 -complex P is said to be minimal for X if X ≻ P and whenever P ≻ Q, Q is homotopy equivalent to P . We also consider special minimal objects called strongly minimal. In this paper we study existence and uniqueness questions. It is an interesting problem to calculate homotopy