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Haynes Robert Miller (born January 29, 1948, in Princeton, New Jersey) [1] is an American mathematician specializing in algebraic topology.. Miller completed his undergraduate study at Harvard University and earned his PhD in 1974 under the supervision of John Coleman Moore at Princeton University with thesis Some Algebraic Aspects of the Adams–Novikov Spectral Sequence. [2]
In mathematics, topological modular forms (tmf) is the name of a spectrum that describes a generalized cohomology theory.In concrete terms, for any integer n there is a topological space , and these spaces are equipped with certain maps between them, so that for any topological space X, one obtains an abelian group structure on the set of homotopy classes of continuous maps from X to .
Trivial topology; Cofinite topology; Finer topology; Product topology. Restricted product; Quotient space; Unit interval; Continuum (topology) Extended real number line; Long line (topology) Sierpinski space; Cantor set, Cantor space, Cantor cube; Space-filling curve; Topologist's sine curve; Uniform norm; Weak topology; Strong topology ...
Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on is allowed to be non-trivial. In, [ 3 ] Sullivan conjectured that η is a weak equivalence after a certain p-completion procedure due to A. Bousfield and D. Kan for the group G = Z / 2 {\displaystyle G=Z/2} .
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its homotopy type. What this looks like is for a space X {\displaystyle X} there is a list of spaces { X n } n ≥ 0 {\displaystyle \{X_{n}\}_{n\geq 0}} where
The Eilenberg–Zilber theorem is a key ingredient in establishing the Künneth theorem, which expresses the homology groups () in terms of () and (). In light of the Eilenberg–Zilber theorem, the content of the Künneth theorem consists in analysing how the homology of the tensor product complex relates to the homologies of the factors.
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes.For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they ...
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group , denoted π 1 ( X ) , {\displaystyle \pi _{1}(X),} which records information about loops in a space .