Algebraic geometry and topology. A symposium in honor of S. by Fox R.H., et al.

By Fox R.H., et al.

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These spaces are generalizations of cells and complexes respectively. Borsuk extended to them numerous theorems valid for the latter. The definition of an ANR space is not exactly local in nature, however it was clear that such spaces are locally smooth in some sense. It was in the presence of these cross-currents that Lefschetz made A space X his contribution to the theory of local-connectedness [57]. locally-connected in dimension q, if for each point x of X, and each neighborhood F of x, there is a neighborhood U of x such that each continuous map of a g-sphere into U can be contracted to a point in F.

If Lv L2 are oriented linear subspaces, of dimensions p, q, a unique L 1 C\L 2 is defined as follows. Select base vectors orientation of x ly xp for L l agreeing with the orientation of L t and so that x n _ q xp is a base for L l n L 2 Complete the latter base to a base x n _ q xn . . , . * , . , L2 . . , , agreeing with its orientation. ,xp is or is not the orientation of L l n L 2 according as x v x n is or is not the orientation for , . . , of the w-space. An orientation of a polyhedral cell

I do not think that it can be claimed that the theory of stacks owes anything to Lefschetz, but, on the other hand, the number of applications which have been made of that theory, particularly to algebraic geometry, owes beyond doubt a very great deal to him. In considering the development of algebraic geometry since 1924 (the date of the publication of the Borel Tract) one is conscious of three main streams first, the classical theory which we owe to the : which the ideas of system of equivalence and canonical systems have emerged; secondly, the abstract algebraic geometry which we owe to Zariski and Weil; and thirdly, the transbeen cendental-topological theory, which has, to a certain extent, Italian school, in merged into the theory of complex manifolds, in which, however, the algebraic varieties keep on distinguishing themselves as the only manifolds on which some of the operations of the theory can be performed.

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