Whitehead torsion
Encyclopedia
In geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...

, the obstruction to a homotopy equivalence  of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion which is an element in the Whitehead group These are named after the mathematician J. H. C. Whitehead
J. H. C. Whitehead
John Henry Constantine Whitehead FRS , known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai , in India, and died in Princeton, New Jersey, in 1960....

.

The Whitehead torsion is important in applying surgery theory
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...

 to non-simply connected manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s of dimension >4: for simply-connected manifolds, the Whitehead group vanishes, and thus homotopy equivalences and simple homotopy equivalences are the same. The applications are to differentiable manifolds, PL manifolds and topological manifolds. The proofs were first
obtained in the early 1960's by Smale, for differentiable manifolds. The development
of handlebody
Handlebody
In the mathematical field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds...

 theory allowed much the same proofs in the differentiable and PL categories.
The proofs are much harder in the topological category, requiring the theory of Kirby
Robion Kirby
Robion Cromwell Kirby is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology...

 and Siebenmann. The restriction to manifolds of dimension >4
are due to the application of the Whitney trick for removing double points.

In generalizing the h-cobordism
H-cobordism
A cobordism W between M and N is an h-cobordism if the inclusion mapsare homotopy equivalences...

 theorem, which is a statement about simply connected manifolds, to non-simply connected manifolds, one must distinguish simple homotopy equivalences and non-simple homotopy equivalences.
While an h-cobordism W between simply-connected closed connected manifolds M and N of dimension n > 4 is isomorphic to a cylinder (the corresponding homotopy equivalence can be taken to be a diffeomorphism, PL-isomorphism, or homeomorphism, respectively), the s-cobordism theorem states that if the manifolds are not simply-connected, an h-cobordism is a cylinder if and only if the Whitehead torsion of the inclusion vanishes.

The Whitehead group

The Whitehead group of a CW-complex or a manifold M is equal to the Whitehead group Wh(π1(M)) of the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

 π1(M) of M.

If G is a group, the Whitehead group Wh(G) is defined to be the cokernel
Cokernel
In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....

 of the map which sends (g,±1) to the invertible (1,1)-matrix (±g). Here Z[G] is the group ring
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...

 of G. Recall that the K-group  of a ring A is defined as the quotient of GL(A) by the subgroup generated by elementary matrices. The group GL(A) is the direct limit
Direct limit
In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...

 of the finite dimensional groups ; concretely, the group of invertible infinite matrices which differ from the identity matrix in only a finite number of coefficients. An elementary matrix here is a transvection
Shear matrix
In mathematics, a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another...

: one such that all main diagonal elements are 1 and there is at most one non-zero element not on the diagonal. The subgroup generated by elementary matrices is exactly the derived subgroup, in other words the smallest normal subgroup such that the quotient by it is abelian.

In other words, the Whitehead group Wh(G) of a group G is the quotient of GL(A) by the subgroup generated by elementary matrices, elements of G and -1. Notice that this is the same as the quotient of the reduced K-group by G.

Examples

  • The Whitehead group of the trivial group
    Trivial group
    In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...

     is trivial. Since the group ring of the trivial group is Z, we have to show that any matrix can be written as a product of elementary matrices times a diagonal matrix; this follows easily from the fact that Z is a Euclidean domain
    Euclidean domain
    In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean algorithm...

    .

  • The Whitehead group of a free abelian group
    Free abelian group
    In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...

     is trivial, a 1964 result of Bass
    Hyman Bass
    Hyman Bass is an American mathematician, known for work in algebra and in mathematics education. From 1959-1998 he was Professor in the Mathematics Department at Columbia University, where he is now professor emeritus...

    , Heller and Swan
    Richard Swan
    Richard Gordon Swan is an American mathematician who is best known for Swan's theorem. His work has mainly been in the area of algebraic K-theory.-External links:**...

    . This is quite hard to prove, but is important as it is used in the proof that an s-cobordism of dimension at least 6 whose ends are tori
    Tori
    Tori may refer to:*Taiwan Ocean Research Institute, the ocean research institute of Taiwan*Tori , horse originating in continental Estonia*Tori , the executor of a technique in partnered martial arts practice...

     is a product. It is also the key algebraic result used in the surgery theory
    Surgery theory
    In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...

     classification of piecewise linear manifolds of dimension at least 5 which are homotopy equivalent to a torus
    Torus
    In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

    ; this is the essential ingredient of the 1969 Kirby-Siebenmann structure theory of topological manifold
    Topological manifold
    In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...

    s of dimension at least 5.

  • The Whitehead group of a braid group
    Braid group
    In mathematics, the braid group on n strands, denoted by Bn, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group...

     (or any subgroup of a braid group) is trivial. This was proved by Farrell and Roushon.

  • The Whitehead group of the cyclic group
    Cyclic group
    In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

    s of orders 2, 3, 4, and 6 are trivial.

  • The Whitehead group of the cyclic group of order 5 is Z. This was proved in 1940 by Higman
    Graham Higman
    Graham Higman FRS was a leading British mathematician. He is known for his contributions to group theory....

    . An example of a non-trivial unit in the group ring is (1−t−t4)(1−t2−t3)=1, where t is a generator of the cyclic group of order 5. This example is closely related to the existence of units of infinite order in the ring of integers of the cyclotomic field generated by fifth roots of unity.

  • The Whitehead group of any finite group G is finitely generated, of rank equal to the number of irreducible real representation
    Real representation
    In the mathematical field of representation theory a real representation is usually a representation on a real vector space U, but it can also mean a representation on a complex vector space V with an invariant real structure, i.e., an antilinear equivariant mapj\colon V\to V\,which...

    s of G minus the number of irreducible rational representation
    Rational representation
    In mathematics, in the representation theory of algebraic groups, a linear representation of an algebraic group is said to be rational if, viewed as a map from the group to the general linear group, it is a rational map of algebraic varieties....

    s. this was proved in 1965 by Bass
    Hyman Bass
    Hyman Bass is an American mathematician, known for work in algebra and in mathematics education. From 1959-1998 he was Professor in the Mathematics Department at Columbia University, where he is now professor emeritus...

    .

  • If G is a finite abelian group then K1(Z[G]) is isomorphic to the units of the group ring Z[G] under the determinant map, so Wh(G) is just the group of units of Z[G] modulo the group of "trivial units" generated by elements of G and −1.

  • It is a well-known conjecture that the Whitehead group of any torsion-free group should vanish.

The Whitehead torsion

At first we define the Whitehead torsion for a chain homotopy equivalence of finite based free R-chain complexes. We can assign to the homotopy equivalence its mapping cone
Mapping cone (homological algebra)
In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that...

 C* := cone*(h*) which is a contractible finite based free R-chain complex. Let be any chain contraction of the mapping cone, i.e. for all . We obtain an isomorphism with , . We define , where A is the matrix of (c* + γ*)odd with respect to the given bases.

For a homotopy equivalence of connected finite CW-complexes we define the Whitehead torsion as follows. Let be the lift of to the universal covering. It induces Z1(Y)]-chain homotopy equivalences . Now we can apply the definition of the Whitehead torsion for a chain homotopy equivalence and obtain an element in which we map to . This is the Whitehead torsion .

Properties

Homotopy invariance: Let be homotopy equivalences of finite connected CW-complexes. If and are homotopic then .

Topological invariance: If is a homeomorphism of finite connected CW-complexes then .

Composition formula: Let , be homotopy equivalences of finite connected CW-complexes. Then .

Geometric interpretation

The s-cobordism theorem states for a closed connected oriented manifold M of dimension > 4 that an h-cobordism
H-cobordism
A cobordism W between M and N is an h-cobordism if the inclusion mapsare homotopy equivalences...

 W between M and another manifold N is trivial over M if and only if the Whitehead torsion of the inclusion vanishes. Moreover, for any element in the Whitehead group there exists an h-cobordism over whose Whitehead torsion is the considered element. The proofs use handlebody decompositions.

There exists a homotopy theoretic analogue of the s-cobordism theorem.
Given a CW-complex A, consider the set of all pairs of CW-complexes (X,A) such that the inclusion of A into X is a homotopy equivalence. Two pairs (X1,A) and (X2,A) are said to be equivalent, if there is a simple homotopy equivalence between X1 and X2 relative to A. The set of such equivalence classes form a group where the addition is given by taking union of X1 and X2 with common subspace A. This group is natural isomorphic to the Whitehead group Wh(A) of the CW-complex A. The proof of this fact is similar to the proof of s-cobordism theorem.

External links

The source of this article is wikipedia, the free encyclopedia.  The text of this article is licensed under the GFDL.
 
x
OK