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Samuel Eilenberg was a Polish-American mathematician who co-founded category theory and homological algebra.
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In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré.
In mathematics, Kuiper's theorem is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible bounded endomorphisms of H is such that all maps from any finite complex Y to GL(H) are homotopic to a constant, for the norm topology on operators.
In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by George W. Whitehead (1942), extending a construction of Heinz Hopf (1935).
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot.
Dennis Parnell Sullivan is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center and is a distinguished professor at Stony Brook University.
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Tudor Ganea was a Romanian-American mathematician, known for his work in algebraic topology, especially homotopy theory. Ganea left Communist Romania to settle in the United States in the early 1960s. He taught at the University of Washington.
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In point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. The answer is 14. This result was first published by Kazimierz Kuratowski in 1922. It gained additional exposure in Kuratowski's fundamental monograph Topologie before achieving fame as a textbook exercise in John L. Kelley's 1955 classic, General Topology.
In mathematics, an acyclic space is a nonempty topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of X are isomorphic to the corresponding homology groups of a point.
Spacetime topology is the topological structure of spacetime, a topic studied primarily in general relativity. This physical theory models gravitation as the curvature of a four dimensional Lorentzian manifold and the concepts of topology thus become important in analysing local as well as global aspects of spacetime. The study of spacetime topology is especially important in physical cosmology.
In mathematics, the Mathai–Quillen formalism is an approach to topological quantum field theory introduced by Atiyah and Jeffrey (1990), based on the Mathai–Quillen form constructed in Mathai and Quillen (1986). In more detail, using the superconnection formalism of Quillen, they obtained a refinement of the Riemann–Roch formula, which links together the Thom classes in K-theory and cohomology, as an equality on the level of differential forms. This has an interpretation in physics as the computation of the classical and quantum (super) partition functions for the fermionic analogue of a harmonic oscillator with source term. In particular, they obtained a nice Gaussian shape representative of the Thom class in cohomology, which has a peak along the zero section.
In topology, a branch of mathematics, given a topological monoid X up to homotopy, an infinite loop space machine produces a group completion of X together with infinite loop space structure. For example, one can take X to be the classifying space of a symmetric monoidal category S; that is, . Then the machine produces the group completion . The space may be described by the K-theory spectrum of S.
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In mathematics, particularly algebraic topology, the Kan-Thurston theorem associates a discrete group to every path-connected topological space in such a way that the group cohomology of is the same as the cohomology of the space . The group might then be regarded as a good approximation to the space , and consequently the theorem is sometimes interpreted to mean that homotopy theory can be viewed as part of group theory.
In algebraic topology, a branch of mathematics, an approximate fibration is a sort of fibration such that the homotopy lifting property holds only approximately. The notion was introduced by Coram and Duvall in 1977.
References
- Steenrod, N.E. (1968). "Milgram's classifying space of a topological group". Topology. 7 (4): 349–368. doi: 10.1016/0040-9383(68)90012-8 . ISSN 0040-9383.
- Fiedorowicz, Z. (1984). "Classifying Spaces of Topological Monoids and Categories". American Journal of Mathematics. 106 (2): 301. doi:10.2307/2374307. ISSN 0002-9327.
