mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a

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 (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dim ...

(''W'', ''M'', ''M''⁻) of an (''n'' + 1)-dimensional

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 ...

(with boundary) ''W'' between its boundary components, two ''n''-manifolds ''M'' and ''M''⁻, is called a semi-''s''-cobordism if (and only if) the inclusion

M \hookrightarrow W

is a

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 ( ...

(as in an ''s''-cobordism), with no further requirement on the inclusion

M^- \hookrightarrow W

(not even being a homotopy equivalence).

Other notations

The original creator of this topic, Jean-Claude Hausmann, used the notation ''M''₋ for the right-hand boundary of the cobordism.

Properties

A consequence of (''W'', ''M'', ''M''⁻) being a semi-''s''-cobordism is that the

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...

of the derived

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

K = \ker(\pi_1(M^) \twoheadrightarrow \pi_1(W))

is perfect. A corollary of this is that

\pi_1(M^)

solves the group extension problem

1 \rightarrow K \rightarrow \pi_1(M^) \rightarrow \pi_1(M) \rightarrow 1

. The solutions to the group extension problem for prescribed

quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...

\pi_1(M)

and kernel group K are classified up to congruence by group cohomology (see Mac Lane's ''Homology'' pp. 124-129), so there are restrictions on which n-manifolds can be the right-hand boundary of a semi-''s''-cobordism with prescribed left-hand boundary M and superperfect kernel group K.

Relationship with Plus cobordisms

Note that if (''W'', ''M'', ''M''⁻) is a semi-''s''-cobordism, then (''W'', ''M''⁻, ''M'') is a plus cobordism. (This justifies the use of ''M''⁻ for the right-hand boundary of a semi-''s''-cobordism, a play on the traditional use of ''M''⁺ for the right-hand boundary of a plus cobordism.) Thus, a semi-''s''-cobordism may be thought of as an inverse to Quillen's Plus construction in the manifold category. Note that (''M''⁻)⁺ must be

diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...

(respectively, piecewise-linearly (PL) homeomorphic) to ''M'' but there may be a variety of choices for (''M''⁺)⁻ for a given closed

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...

(respectively, PL) manifold ''M''.

References

* *. *. *{{citation, url=https://doi.org/10.1007%2FBF02566068 , first=Jean-Claude , last=Hausmann , title=Manifolds with a Given Homology and Fundamental Group , journal=

Commentarii Mathematici Helvetici The ''Commentarii Mathematici Helvetici'' is a quarterly peer-reviewed scientific journal in mathematics. The Swiss Mathematical Society started the journal in 1929 after a meeting in May of the previous year. The Swiss Mathematical Society sti ...

, volume=53 , issue=1 , year=1978 , pages=113–134, doi=10.1007/BF02566068. Manifolds Geometric topology Algebraic topology Homotopy theory