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A stably free nonfree module and its relevance for homotopy classification, case \(Q_28\). (English) Zbl 1084.57003
The result of the present paper grew out of joint work of the authors with P. Latiolais on homotopy types of 2-complexes with a finite 3-manifold group as a fundamental group [Proc. Edinb. Math. Soc., II. Ser. 40, No. 1, 69–84 (1997; Zbl 0927.57003)]. Certain stably free but nonfree projective modules detected by Swan give rise to the question whether these can be realized by the second homotopy of two-dimensional CW-complexes thus yielding exotic homotopy types.
Moreover, the outcome is crucial for the still open D(2)-problem raised by C. T. C. Wall. The authors present a very important intermediate step. For \(G\) being the generalized quaternion group of order 28, they give a concrete realization of Swan’s module in a \(Z(G)\)-complex that algebraically looks like the universal covering complex of a presentation of \(G\). The explicitly given low-size matrices give the best hope to decide, whether the chain homotopy type of this realization actually is geometric or not.

MSC:
57M20 Two-dimensional complexes (manifolds) (MSC2010)
55P15 Classification of homotopy type
19A13 Stability for projective modules
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References:
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