# Exact Sequences $\sum_p (K, L)$ and their Applications

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Journal of the Institute of Polytechnics^ Osaka City University, Vol, 3, No, 1—2, Series A E xact S equ en ces {Ky L) a n d th e ir A p p lic a tio n s By Minoru N a k a o k a (Received September 15, 1952) § 0. Introduction Homotopy classificatioiis of mappings of an n dimensional finite cell complex into an ^-sphere or an (;?-l)-sphere and the corresponding extension theorems were solved by H. Hopf and N. E. Steenrod [5] respectively. Intro ducing the cohomotopy group, E. Spanier [4] unified these results in an exact sequence, while J. H. C. Whitehead [9] gave a general and constructive method to obtain an exact sequence, starting with a certain sequence of homomorphisms. In this paper, we shall define exact sequences Sp by applying White head's method to the cohomotopy group of a complex i f (§1). It is proved that Ylp QK') are invariances of homotopy type of complex K (§2), and that, as its special case. So ( i f ) may be regarded as a generalization of Spanier's sequence (^3, 4,5). Si? ( i f ) are also utiUzed to obtain a homotopy classification theorem and a corresponding extension theorem concerning mappings of a certain kind of an (72+2)-dimensional complex into S” (§6). Furthermore we determine the ^-th cohomotopy group of an A^-polyhedron in terms of its cohomology system (g7 ). At the end of this paper it is shown that two A^-polyhedra are of the same homotopy type if and only if their Spanier's sequences are properly isomorphic. I am deeply grateful to Prof. A. Komatu and Mr. H. Uehara for their kind advices during the preparation of this paper. g I. Exact sequences Si?(^> *^) In the first place, let us define an exact sequence S abstractly, following J. H. C. Whitehead [9]. Let r be an arbitrary fixed integer, and let (C, A ) be the following sequence of groups and homomorphisms; p r - i j r jO jq-¥\ (1 ,1 ) —> A " ...... ..................................................................., where C®, are arbitrary abelian groups

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