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Register a new userThe homotopy type of a finite 2-complex with non-minimal Euler characteristic
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John Nicholson
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We resolve two long-standing and closely related problems concerning stably free $mathbb{Z} G$-modules and the homotopy type of finite 2-complexes. In particular, for all $k ge 1$, we show that there exists a group $G$ and a non-free stably free $mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k ge 0$, there exists homotopically distinct finite 2-complexes with fundamental group $G$ and with Euler characteristic $k$ greater than the minimal value over $G$. This provides a solution to Problem D5 in the 1979 Problems List of C. T. C. Wall.
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Baez asks whether the Euler characteristic (defined for spaces with finite homology) can be reconciled with the homotopy cardinality (defined for spaces with finite homotopy). We consider the smallest infinity category $text{Top}^text{rx}$ containing both these classes of spaces and closed under homotopy pushout squares. In our main result, we compute the K-theory $K_0(text{Top}^text{rx})$, which is freely generated by equivalence classes of connected p-finite spaces, as p ranges over all primes. This provides a negative answer to Baezs question globally, but a positive answer when we restrict attention to a prime.
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Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homotopy coincides with strong shape. We attempt to simplify foundations of the theory and to clarify and improve some of its major results. Using geometric tools such as Milnors telescope compactification, comanifolds (=mock bundles) and the Pontryagin-Thom Construction, we obtain new simple proofs of results by Barratt-Milnor; Cathey; Dydak-Segal; Eda-Kawamura; Edwards-Geoghegan; Fox; Geoghegan-Krasinkiewicz; Jussila; Krasinkiewicz-Minc; Mardesic; Mittag-Leffler/Bourbaki; and of three unpublished results by Shchepin. An error in Lisitsas proof of the Hurewicz theorem in Steenrod homotopy is corrected. It is shown that over compacta, R.H.Foxs overlayings are same as I.M.James uniform covering maps. Other results include: - A morphism between inverse sequences of countable (possibly non-abelian) groups that induces isomorphisms on inverse and derived limits is invertible in the pro-category. This implies the Whitehead theorem in Steenrod homotopy, thereby answering two questions of A.Koyama. - If X is an LC_{n-1} compactum, n>0, its n-dimensional Steenrod homotopy classes are representable by maps S^nto X, provided that X is simply connected. The assumption of simply-connectedness cannot be dropped by a well-known example of Dydak and Zdravkovska. - A connected compactum is Steenrod connected (=pointed 1-movable) iff every its uniform covering space has countably many uniform connected components.
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