Whitehead torsion
Encyclopedia
In geometric topology
, 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
.
The Whitehead torsion is important in applying surgery theory
to non-simply connected 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
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
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
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.
π1(M) of M.
If G is a group, the Whitehead group Wh(G) is defined to be the cokernel
of the map
which sends (g,±1) to the invertible (1,1)-matrix (±g). Here Z[G] is the group ring
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
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
: 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.
for a chain homotopy equivalence 
of finite based free R-chain complexes. We can assign to the homotopy equivalence its mapping cone
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 Z[π1(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 
.
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 
.
> 4 that an h-cobordism
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.
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. WhiteheadJ. 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 groupFundamental 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 ringGroup 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 limitDirect 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 transvectionShear 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 groupTrivial groupIn 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 domainEuclidean domainIn 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 groupFree abelian groupIn 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 BassHyman BassHyman 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 SwanRichard SwanRichard 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 toriToriTori 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 theorySurgery theoryIn 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 torusTorusIn 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 manifoldTopological manifoldIn 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 groupBraid groupIn 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 groupCyclic groupIn 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 HigmanGraham HigmanGraham 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 representationReal representationIn 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 representationRational representationIn 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 BassHyman BassHyman 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 coneMapping 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 Z[π1(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-cobordismH-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.