Homotopy Theory. The Mathematical Works of J. H. C. by John Henry Constantine Whitehead

By John Henry Constantine Whitehead

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E. a homeomorph of σ η ), which has no point in common with K. Let / : En -> Κη~χ be any map of En} the boundary of En, into if"" 1 . Let en == En - En and let φ : K U En -> K U e n be the map which is given by φ \ K\Jen = 1, φ p = f p if p ε En. Let UT + e n be the space which consists of the points in K U e n with the identification topology determined by φ. That is to say a set XaK -\- en is closed if, and only if, φ^ΧαΚϋΕ11 is closed. e. t h a t K and en retain their topologies in K + en. Therefore K -f en is a complex, whose cells are the cells in K, together with en.

Let this be so and let / ε x> gey. Let a — (m,r), b = (m,s), d = (m,r,s) = (m,c). Then (m/a) / εμ,^^χ, (m/6) gr ε ^ ^ , whence m2 - ^ f g ε μτηίΤ (x) μτη,* (y) Also (m/dj/flre/^isy) and mei m . 3) are represented by the same co-cycle, " ) See [8], [9], [10], [11] and [6], chap. V. 3) is established. 2) if m = c = (r, s). For then m = (m, r) = (m, $) = (m, r, s) and // m e = 1. 4) where x ε H*>(r), y ε R(s), c = (r, s) and xe = μ ^ ζ , j/ c = //Cj,t/. 2), where / eCp, g eCq. However we shall not need this because of the special nature of our complexes.

H. Acad, Sc. (Doklady), 34,'i p. 35-37· N. K. , 48, 1947. M. BOCKSTEIN, C R. Acad. Sc. (Doklady), 37, 19^2, p. 243-245. J. H. ('. , 42, 194»» P· 409~42^· J. H. C. WIIITEHEAD, Proc. L. M. , 45, 1939, p. 243-327. ON SIMPLY CONNECTED, 4-D1MENSIONAL POLYHEDRA Received January 17, 1948 1. Introduction. Our main purpose is to show that the homotopy type of a simply connected, 4-dimensional polyhedron is completely de­ termined by its inter-related co-homology rings, mod. m (m = 0, 1, 2 , . . ), together with one additional element of structure.

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