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In mathematics, a normal map is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex X (more geometrically a Poincaré space), a normal map on X endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, X has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold M to X matching the fundamental classes and preserving normal bundle information. If the dimension of X is 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to X actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by Sergei Novikov.
The cobordism classes of normal maps on X are called normal invariants. Depending on the category of manifolds (differentiable, piecewise-linear, or topological), there are similarly defined, but inequivalent, concepts of normal maps and normal invariants.
It is possible to perform surgery on normal maps, meaning surgery on the domain manifold, and preserving the map. Surgery on normal maps allows one to systematically kill elements in the relative homotopy groups by representing them as embeddings with trivial normal bundle.
defined, but inequivalent, concepts of normal maps and normalinvariants. It is possible to perform surgery on normal maps, meaning surgery on the domain...
abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by...
Normal map may refer to: Normal mapping in 3D computer graphics Normalinvariants in mathematical surgery theory Normal matrix in linear algebra Normal...
functions Normal function, in set theory Normalinvariants, in geometric topology Normal matrix, a matrix that commutes with its conjugate transpose Normal measure...
tests are affine invariant but not consistent. For example, the multivariate skewness test is not consistent against symmetric non-normal alternatives. The...
divisors used in the construction of the Jordan normal form do not exist over F[X], so the invariant factors fi as given above must be used instead. The...
matrix A may be put in Jordan normal form. Since the underlying vector space can be shown to be the direct sum of invariant subspaces associated with the...
which are usually easier to determine. These are on one hand the normalinvariants which form generalized cohomology groups, and hence one can use standard...
orientability (a normalinvariant, also detected by homology) and genus (a homological invariant). Smooth closed manifolds have no local invariants (other than...
closely related concept is self-similarity, where a function or curve is invariant under a discrete subset of the dilations. It is also possible for the...
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician Cahit Arf (1941)...
equation Elementary divisors Frobenius normal form (also called Rational canonical form) Hermite normal form Invariant factor Structure theorem for finitely...
(called normalinvariants) are classified by the set of homotopy classes [ X , G / O ] {\displaystyle [X,G/O]} . Each of these normalinvariants has a surgery...
mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological...
characteristic in G. A subgroup of H that is invariant under all inner automorphisms is called normal; also, an invariant subgroup. ∀φ ∈ Inn(G)： φ[H] ≤ H Since...
addition each Ai is normal in G, then the series is called a normal series, when this term is not used for the weaker sense, or an invariant series. A series...
operator is called normal if the space admits a decomposition into a direct sum of a finite-dimensional generalized eigenspace and an invariant subspace where...
eigenvalues. The invariant subspaces of a shift acting on Hardy space are characterized by Beurling's theorem. The notion of normal operators generalizes...
In mathematics, the Kervaire invariant is an invariant of a framed ( 4 k + 2 ) {\displaystyle (4k+2)} -dimensional manifold that measures whether the...
mathematics, a group is supersolvable (or supersoluble) if it has an invariantnormal series where all the factors are cyclic groups. Supersolvability is...
genus) and orientability are a complete set of invariants. Jordan normal form of a matrix is a complete invariant for matrices up to conjugation, but eigenvalues...
non-invariant fluids, such as polymers. At every point in a stressed body there are at least three planes, called principal planes, with normal vectors...
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical...
{\textstyle \kappa _{n}(X+c)=\kappa _{n}(X),} i.e. the cumulant is translation invariant. (If n = 1 {\textstyle n=1} then we have κ 1 ( X + c ) = κ 1 ( X ) + c...
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded...
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