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geometric topology In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topo ...
, a field within mathematics, the obstruction to a
homotopy equivalence In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
f\colon X \to Y of finite
CW-complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
es being a simple homotopy equivalence is its Whitehead torsion \tau(f) which is an element in the Whitehead group \operatorname(\pi_1(Y)). These concepts are named after the mathematician J. H. C. Whitehead. The Whitehead torsion is important in applying
surgery theory In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while An ...
to non-
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
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 1960s by
Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty ...
, for differentiable manifolds. The development of
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. Handles ...
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
Robion Kirby Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology. Together with Laurent C. Siebenmann he developed the Kirby–Siebenmann invariant ...
and Laurent C. Siebenmann. The restriction to manifolds of dimension greater than four 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 In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
the Whitehead torsion of the inclusion M \hookrightarrow W vanishes.


Whitehead group

The Whitehead group of a connected CW-complex or a manifold ''M'' is equal to the Whitehead group \operatorname(\pi_1(M)) of the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
\pi_1(M) of ''M''. If ''G'' is a group, the Whitehead group \operatorname(G) is defined to be the
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...
of the map G\times \ \to K_1(\Z which sends (''g'', ±1) to the invertible (1,1)-matrix (±''g''). Here \Z /math> is the
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 the set of elements of the gi ...
of ''G''. Recall that the K-group K1(''A'') 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 In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any cate ...
of the finite-dimensional groups GL(''n'', ''A'') → GL(''n''+1, ''A''); concretely, the group of invertible infinite matrices which differ from the identity matrix in only a finite number of coefficients. An
elementary matrix In mathematics, an elementary matrix is a square matrix obtained from the application of a single elementary row operation to the identity matrix. The elementary matrices generate the general linear group when is a field. Left multiplication (p ...
here is a transvection: one such that all
main diagonal In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix ...
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 mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal s ...
, in other words the smallest
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
such that the quotient by it is abelian. In other words, the Whitehead group \operatorname(G) of a group ''G'' is the quotient of \operatorname(\Z by the subgroup generated by elementary matrices, elements of ''G'' and \pm 1. Notice that this is the same as the quotient of the reduced K-group \tilde_1(\Z by ''G''.


Examples

*The Whitehead group of the
trivial group In mathematics, a trivial group or zero group is a group that consists 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 and so it is usu ...
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 In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of Euclidean division of integers. Th ...
. *The Whitehead group of a
free abelian group In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation (mathematics), operation that is associative, commutative, and inverti ...
is trivial, a 1964 result of Hyman Bass, Alex Heller and
Richard Swan Richard Gordon Swan (; born 1933) is an American mathematician who is known for the Serre–Swan theorem relating the geometric notion of vector bundles to the algebraic concept of projective modules, and for the Swan representation, an ''l''-a ...
. 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 is a product. It is also the key algebraic result used in the
surgery theory In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while An ...
classification of piecewise linear
manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
of dimension at least 5 which are homotopy equivalent to a
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
; this is the essential ingredient of the 1969 Kirby–Siebenmann structure theory of
topological manifold In topology, a topological manifold is a topological space that locally resembles real ''n''- dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds ...
s of dimension at least 5. *The Whitehead group of a
braid group In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of Braid theory, -braids (e.g. under ambient isotopy), and whose group operation is composition of ...
(or any subgroup of a braid group) is trivial. This was proved by F. Thomas Farrell and Sayed K. Roushon. *The Whitehead group of a finite
cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
of order n is trivial if and only if n is 1, 2, 3, 4, or 6. *The Whitehead group of the cyclic group of order 5 is \Z. This was proved in 1940 by
Graham Higman Graham Higman FRS (19 January 1917 – 8 April 2008) was a prominent English mathematician known for his contributions to group theory. Biography Higman was born in Louth, Lincolnshire, and attended Sutton High School, Plymouth, winning ...
. An example of a non-trivial unit in the group ring arises from the identity (1-t-t^4)(1-t^2-t^3)=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 particular, the
golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...
) in the ring of integers of the
cyclotomic field In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory ...
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 representations of ''G'' minus the number of irreducible
rational representation In mathematics, in the representation theory of algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebrai ...
s. This was proved in 1965 by Bass. * If ''G'' is a finite cyclic group then K_1(\Z is isomorphic to the units of the group ring \Z /math> under the determinant map, so Wh(''G'') is just the group of units of \Z /math> 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 \tau(h_*) \in _1(R) for a chain homotopy equivalence h_*: D_* \to E_* of finite based free ''R''-chain complexes. We can assign to the homotopy equivalence its
mapping cone Mapping cone may refer to one of the following two different but related concepts in mathematics: * Mapping cone (topology) * Mapping cone (homological algebra) {{mathdab ...
C* := cone*(h*) which is a contractible finite based free ''R''-chain complex. Let \gamma_*: C_* \to C_ be any chain contraction of the mapping cone, i.e., c_ \circ \gamma_n + \gamma_ \circ c_n = \operatorname_ for all ''n''. We obtain an isomorphism (c_* + \gamma_*)_\mathrm: C_\mathrm \to C_\mathrm with :C_\mathrm := \bigoplus_ C_n,\qquad C_\mathrm := \bigoplus_ C_n. We define \tau(h_*) := \in _1(R), where ''A'' is the matrix of (c_* + \gamma_*)_ with respect to the given bases. For a homotopy equivalence f: X\to Y of connected finite CW-complexes we define the Whitehead torsion \tau(f) \in \operatorname(\pi_1(Y)) as follows. Let : \to be the lift of f: X \to Y to the universal covering. It induces \Z pi_1(Y)/math>-chain homotopy equivalences C_*(): C_*() \to C_*(). Now we can apply the definition of the Whitehead torsion for a chain homotopy equivalence and obtain an element in _1(\Z pi_1(Y) which we map to Wh(π1(''Y'')). This is the Whitehead torsion τ(ƒ) ∈ Wh(π1(''Y'')).


Properties

Homotopy invariance: Let f,g\colon X\to Y be homotopy equivalences of finite connected CW-complexes. If ''f'' and ''g'' are homotopic, then \tau(f) = \tau(g). Topological invariance: If f\colon X\to Y is a homeomorphism of finite connected CW-complexes, then \tau(f) = 0. Composition formula: Let f\colon X\to Y, g\colon Y\to Z be homotopy equivalences of finite connected CW-complexes. Then \tau(g \circ f) = g_* \tau(f) + \tau(g).


Geometric interpretation

The s-cobordism theorem states for a closed connected oriented manifold ''M'' of dimension ''n'' > 4 that an
h-cobordism In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : ...
''W'' between ''M'' and another manifold ''N'' is trivial over ''M'' if and only if the Whitehead torsion of the inclusion M\hookrightarrow W vanishes. Moreover, for any element in the Whitehead group there exists an h-cobordism ''W'' over ''M'' whose Whitehead torsion is the considered element. The proofs use
handle decomposition In mathematics, a handle decomposition of an ''m''-manifold ''M'' is a union \emptyset = M_ \subset M_0 \subset M_1 \subset M_2 \subset \dots \subset M_ \subset M_m = M where each M_i is obtained from M_ by the attaching of i-handles. A handle dec ...
s. There exists a homotopy theoretic analogue of the s-cobordism theorem. Given a
CW-complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
''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 (''X''1, ''A'') and (''X''2, ''A'') are said to be equivalent, if there is a simple homotopy equivalence between ''X''1 and ''X''2 relative to ''A''. The set of such equivalence classes form a group where the addition is given by taking union of ''X''1 and ''X''2 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.


See also

*
Algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sens ...
* Reidemeister torsion * s-Cobordism theorem * Wall's finiteness obstruction


References

* *Cohen, M. ''A course in simple homotopy theory'' Graduate Text in Mathematics 10, Springer, 1973 * * * * *{{citation, first=J. H. C., last= Whitehead, authorlink=J. H. C. Whitehead, title=Simple homotopy types, journal=
American Journal of Mathematics The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United S ...
, volume= 72, year= 1950, issue= 1, pages= 1–57, mr=0035437, doi=10.2307/2372133, jstor= 2372133


External links


A description of Whitehead torsion is in section two
Geometric topology Algebraic K-theory Surgery theory