H-cobordism
Encyclopedia
A cobordism
W between M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps
are homotopy equivalences. If
and
, it is called an 
-dimensional h-cobordism.
The h-cobordism theorem states that if:
then W is Cat-isomorphic to M × [0, 1] and (hence) M is Cat-isomorphic to N.
Informally, "a simply connected h-cobordism is a cylinder".
The theorem was first proved by Stephen Smale
and is the fundamental result in the theory of high-dimensional manifolds: for a start, it almost immediately proves the Generalized Poincaré Conjecture
.
The theorem is still true topologically but not smoothly for n = 4; a later result of Michael Freedman
.
Before Smale proved this theorem, mathematicians had got stuck trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were even harder. The h-cobordism theorem showed that (simply connected) manifolds of dimension at least 5 are much easier than those of dimension 3 or 4. The proof of the theorem depends on the "Whitney trick" of Hassler Whitney
, which geometrically untangles homologically-tangled spheres of complementary dimension in a manifold of dimension >5. An informal reason why manifolds of dimension 3 or 4 are unusually hard is that the trick fails to work in lower dimensions, which have no room for untanglement, and so have more tangles.
using a 4-dimensional Whitney trick) but is false PL and smoothly (as shown by Simon Donaldson
).
For n = 3, the h-cobordism theorem for smooth manifolds has not been proved and, due to the Poincaré conjecture, is equivalent to the hard open question of whether the 4-sphere has non-standard smooth structure
s.
For n = 2, the h-cobordism theorem is true – it is equivalent to the Poincaré conjecture
, which has been proved by Grigori Perelman
.
For n = 1, h-cobordism theorem is vacuously true, since there is no closed simply-connected 1-dimensional manifold.
For n = 0, the h-cobordism theorem is trivially true: the interval is the only connected cobordism between connected 0-manifolds.
τ (W, M) of the inclusion
.
Precisely, the s-cobordism theorem (the s stands for simple-homotopy equivalence
), proved independently by Barry Mazur
, John Stallings, and Dennis Barden
, states (assumptions as above but where M and N need not be simply connected):
The torsion vanishes if and only if the inclusion
is not just a homotopy equivalence, but a simple homotopy equivalence.
Note that one need not assume that the other inclusion
is also a simple homotopy equivalence—that follows from the theorem.
Categorically, h-cobordisms form a groupoid
.
Then a finer statement of the s-cobordism theorem is that the isomorphism classes of this category (up to Cat-isomorphism of h-cobordisms) are torsors for the respective
Whitehead group
s
, where 
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
W between M and N is an h-cobordism (the h stands for homotopy equivalence) if the inclusion maps
are homotopy equivalences. If
and
, it is called an -dimensional h-cobordism.The h-cobordism theorem states that if:
- W is a compact h-cobordism between M and N
- in the category Cat=Diff, PLPiecewise linear manifoldIn mathematics, a piecewise linear manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions.An isomorphism of PL manifolds is called a PL...
, or TopTopological manifoldIn mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below... - M and N are simply connected
- dimensionDimensionIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
M and N > 4
then W is Cat-isomorphic to M × [0, 1] and (hence) M is Cat-isomorphic to N.
Informally, "a simply connected h-cobordism is a cylinder".
The theorem was first proved by Stephen Smale
Stephen Smale
Steven Smale a.k.a. Steve Smale, Stephen Smale is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley .-Education and career:He entered the University of...
and is the fundamental result in the theory of high-dimensional manifolds: for a start, it almost immediately proves the Generalized Poincaré Conjecture
Generalized Poincaré conjecture
In the mathematical area of topology, the term Generalized Poincaré conjecture refers to a statement that a manifold which is a homotopy sphere 'is' a sphere. More precisely, one fixes a...
.
The theorem is still true topologically but not smoothly for n = 4; a later result of Michael Freedman
Michael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
.
Before Smale proved this theorem, mathematicians had got stuck trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were even harder. The h-cobordism theorem showed that (simply connected) manifolds of dimension at least 5 are much easier than those of dimension 3 or 4. The proof of the theorem depends on the "Whitney trick" of Hassler Whitney
Hassler Whitney
Hassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
, which geometrically untangles homologically-tangled spheres of complementary dimension in a manifold of dimension >5. An informal reason why manifolds of dimension 3 or 4 are unusually hard is that the trick fails to work in lower dimensions, which have no room for untanglement, and so have more tangles.
Low dimensions
For n = 4, the h-cobordism theorem is true topologically (proved by Michael FreedmanMichael Freedman
Michael Hartley Freedman is a mathematician at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the Poincaré conjecture. Freedman and Robion Kirby showed that an exotic R4 manifold exists.Freedman was born...
using a 4-dimensional Whitney trick) but is false PL and smoothly (as shown by Simon Donaldson
Simon Donaldson
Simon Kirwan Donaldson FRS , is an English mathematician known for his work on the topology of smooth four-dimensional manifolds. He is now Royal Society research professor in Pure Mathematics and President of the Institute for Mathematical Science at Imperial College London...
).
For n = 3, the h-cobordism theorem for smooth manifolds has not been proved and, due to the Poincaré conjecture, is equivalent to the hard open question of whether the 4-sphere has non-standard smooth structure
Smooth structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....
s.
For n = 2, the h-cobordism theorem is true – it is equivalent to the Poincaré conjecture
Poincaré conjecture
In mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
, which has been proved by Grigori Perelman
Grigori Perelman
Grigori Yakovlevich Perelman is a Russian mathematician who has made landmark contributions to Riemannian geometry and geometric topology.In 1992, Perelman proved the soul conjecture. In 2002, he proved Thurston's geometrization conjecture...
.
For n = 1, h-cobordism theorem is vacuously true, since there is no closed simply-connected 1-dimensional manifold.
For n = 0, the h-cobordism theorem is trivially true: the interval is the only connected cobordism between connected 0-manifolds.
The s-cobordism theorem
If the assumption that M and N are simply connected is dropped, h-cobordisms need not be cylinders; the obstruction is exactly the Whitehead torsionWhitehead torsion
In geometric topology, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau, which is an element in the Whitehead group Wh. These are named after the mathematician J. H. C...
τ (W, M) of the inclusion
.Precisely, the s-cobordism theorem (the s stands for simple-homotopy equivalence
Simple-homotopy equivalence
In mathematics, particularly the area of topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they are related by a sequence of collapses and expansions , and a homotopy equivalence is a simple homotopy...
), proved independently by Barry Mazur
Barry Mazur
-Life:Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate school and received his Ph.D. from Princeton University in 1959,...
, John Stallings, and Dennis Barden
Dennis Barden
Dennis Barden is a mathematician at the University of Cambridge, who proved the s-cobordism theorem and classified the simply connected compact 5-manifolds.-Publications:...
, states (assumptions as above but where M and N need not be simply connected):
- an h-cobordism is a cylinder if and only if Whitehead torsionWhitehead torsionIn geometric topology, the obstruction to a homotopy equivalence f\colon X \to Y of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion \tau, which is an element in the Whitehead group Wh. These are named after the mathematician J. H. C...
τ (W, M) vanishes
The torsion vanishes if and only if the inclusion
is not just a homotopy equivalence, but a simple homotopy equivalence.Note that one need not assume that the other inclusion
is also a simple homotopy equivalence—that follows from the theorem.Categorically, h-cobordisms form a groupoid
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...
.
Then a finer statement of the s-cobordism theorem is that the isomorphism classes of this category (up to Cat-isomorphism of h-cobordisms) are torsors for the respective
Whitehead group
Whitehead group
Whitehead group in mathematics may mean:* A group W with Ext=0; see Whitehead problem* For a ring, the Whitehead group Wh of a ring A, equal to K_1...
s
, where