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Solomon Lefschetz ForMemRS (Russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. [3] [1] [4] [5]
This terminology is often used in the case of the algebraic topology on the set of discrete, faithful representations of a Kleinian group into PSL(2,C). Another topology, the geometric topology (also called the Chabauty topology), can be put on the set of images of the representations, and its closure can include extra Kleinian groups that are ...
A space of all maps from to X, with no distinguished point for the start of the paths, is called the free path space of X. [2] The maps from to X are called free paths. The path space P X {\displaystyle PX} is then the pullback of X I → X , χ ↦ χ ( 0 ) {\displaystyle X^{I}\to X,\,\chi \mapsto \chi (0)} along ∗ ↪ X {\displaystyle ...
Note that if this direct map () () of cochain complexes were in fact a map of differential graded algebras, then the cup product would make () a commutative graded algebra, which it is not. This failure of the Alexander–Whitney map to be a coalgebra map is an example the unavailability of commutative cochain-level models for cohomology over ...
A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406
While this kind of topology was still "combinatorial" in 1942, it had become "algebraic" by 1944. [4] This corresponds also to the period where homological algebra and category theory were introduced for the study of topological spaces, and largely supplanted combinatorial methods.
Download as PDF; Printable version; ... This page was last edited on 30 October 2023, at 12:17 ... List of algebraic topology topics.
Differential graded algebra: the algebraic structure arising on the cochain level for the cup product; Poincaré duality: swaps some of these; Intersection theory: for a similar theory in algebraic geometry