By Solomon Lefschetz

Because the booklet of Lefschetz's Topology (Amer. Math. Soc. Colloquium guides, vol. 12, 1930; stated less than as (L)) 3 significant advances have motivated algebraic topology: the improvement of an summary advanced self reliant of the geometric simplex, the Pontrjagin duality theorem for abelian topological teams, and the tactic of Cech for treating the homology conception of topological areas by way of structures of "nerves" every one of that is an summary advanced. the result of (L), very materially additional to either by means of incorporation of next released paintings and by means of new theorems of the author's, are right here thoroughly recast and unified by way of those new suggestions. A excessive measure of generality is postulated from the outset.

The summary viewpoint with its concomitant formalism allows succinct, distinctive presentation of definitions and proofs. Examples are sparingly given, usually of an easy type, which, as they don't partake of the scope of the corresponding textual content, will be intelligible to an trouble-free pupil. yet this is often essentially a ebook for the mature reader, within which he can locate the theorems of algebraic topology welded right into a logically coherent complete

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**Example text**

Then

R is divided accordingly into subsets <€(e) and it turns out that, for each e , all curves (e) * in <£ have the same analytic type. 3. Suppose 2k is a cubefree integer and let D the set of boundary curves in (1) # j0) where, for each e , all boundary curves C e # (e) are isomorphic to the modular curve Y,(e) (2) (e) — Suppose C e # , let C denote the closure of D. /r. , and consider the commutative diagram Yx(e) = + a C* c o.

5) hence can be detected in the rational closure of the cone C(F) . The main problem in this case consists in finding explicit r,-admissible decompositions of the cones C(F) and in understanding the geometric meaning of the associated toroidal compactifications. To this end, one might be able to use certain results of Vinberg that we now briefly describe. These results can be applied towards the construction of fundamental polyhedra for the action of on 0(E /E) C(F) , which should yield admissible decompositions, and towards the explicit determination of the rational closure of examples.