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Associativity of the external cross-product (for three complexes) entirely obvious. The commutation law is easily derived. Let TiKxK'^K'xKbe defined by T(z, x = (x f ) onto cells; hence it f , x). induces a unique chain map. Then A T maps cells simple induction based on dT = Td yields where p, q are the dimensions of cr, T. This gives immediately T(u xv) = (-l)wvxu. 4) Thus the external cross-products are easily and uniquely defined, and all properties are readily derivable. Now comes the essential feature of Lefschetz's method: the derivation of the cohomology cup- K K K ->K x be a complex, let d product from the cross-product.

Sbornik), 39, no. 3: 97-102. Rec. 1933 [54] On On [55] On [53] singular chains and cycles. Bull. Amer. Math. , 39: 124-129. analytical complexes (with J. H. C. Whitehead). Trans. Amer. Math. , 35: 510-517. Amer. generalized manifolds. J. , 55: 469-504. 1934 Elementary One- and Two -Dimensional Topology (a course given by Prof. Lefschetz, Spring 1934 notes by H. Wallman). Princeton University. , mimeographed. [57] On locally connected and related sets. Ann. of Math. (2), 35: 118-129. [56] ; 1935 [58] Topology (lectures 1934-35; notes by N.

In this case a simple induction K , LEFSCHBTZ AND ALGEBRAIC TOPOLOGY 29 shows that if the closed cells (T, f have a non-vacuous intersection, then or n T is an open convex set in L(cr) n L(r) whose boundary is a polyhedron whose cells are the intersections of cr and its faces with r and n faces (excluding cr T). orientation of a vector space is an equivalence class of sets of basis vectors, two bases being equivalent if the matrix relating the its An two bases has a positive determinant. There are thus two orientations, and an orientation is specified by a set of basis vectors.

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