A geometric approach to homology theory.

*(English)*Zbl 0315.55002
London Mathematical Society Lecture Note Series. 18. Cambridge etc.: Cambridge University Press. 149 p. £3.90 (1976).

Summary: From the Introduction: “The purpose of these notes is to give a geometrical treatment of generalised homology and cohomology theories. The central idea is that of a ‘mock bundle’, which is the geometric cocycle of a general cobordism theory, and the main new result is that any homology theory is a generalised bordism theory. Thus every theory has both cycles and cocycles; the cycles are manifolds, with a pattern of singularities depending on the theory, and the cocycles are mock bundles with the same ‘manifolds’ as fibres.

The geometric treatment, which we give in detail for the case of PL-bordism and cobordism, has many good features. Mock bundles are easy to set up and to see as a cohomology theory. Duality theorems are transparent (the PoincarĂ© duality map is the identity on representatives). Thom isomorphism and the cohomology transfer are obvious geometrically while cup product is just ‘Whitney sum’ on the bundle level and cap product is the induced bundle glued up. Transversality is built into the theory – the geometric interpretations of cup and cap products are extensions of those familiar in classical homology. Coefficients have a beautiful geometrical interpretation and the universal coefficient sequence is absorbed into the more general ‘killing’ exact sequence. Equivariant cohomology is easy to set up and operations are defined in a general setting. Finally there is the new concept of a generalised cohomology with a sheaf of coefficients (which unfortunately does not have all the nicest properties).”

These notes will provide matter for thought for all algebraic topologists, particularly those who were brought up to regard PL-theories as being difficult (since there is no Lie group corresponding to PL-manifolds in the same way that, e.g. \(O(n)\) corresponds to smooth unoriented manifolds.)

After a reasonable amount of preparation, the authors introduce mock bundles. A mock bundle over a ball complex \(K\) consists of a PL-projection \(E\overset {p}\rightarrow\vert K\vert\) such that for each \(\sigma\in K\), \(p^{-1}(\sigma)\) is a compact PL manifold of dimension \(q+\dim(\sigma)\) with boundary \(p^{-1}(\dot\sigma)\). Think of these as bundles (for functoriality) but impose a cobordism equivalence relation. One obtains the PL-cobordism theory.

Generalizations of this basic idea together with a related study of singularities of manifolds form the core of these notes.

The reviewer believes that the authors’ evident enthusiasm for their approach is justified. Particularly striking is the treatment of the transfer, the discussion of \(Z_2\) operations on PL-cobordism, and the fact that any CW spectrum may be viewed as the Thom spectrum of a suitable bordism theory.

The authors dedicate their work to Dennis Sullivan – indeed, these notes will serve to propagate his ideas.

The notes are not self-contained, but the authors carefully provide references for missing proofs. Some familiarity with PL-topology seems essential. Sophisticated ideas are treated on occasion in a deceptively simple manner. The notes are physically attractive with many line illustrations. They might be suitable for a graduate level seminar or for the mathematician who is simply curious to see how geometry is currently invading topology.

The geometric treatment, which we give in detail for the case of PL-bordism and cobordism, has many good features. Mock bundles are easy to set up and to see as a cohomology theory. Duality theorems are transparent (the PoincarĂ© duality map is the identity on representatives). Thom isomorphism and the cohomology transfer are obvious geometrically while cup product is just ‘Whitney sum’ on the bundle level and cap product is the induced bundle glued up. Transversality is built into the theory – the geometric interpretations of cup and cap products are extensions of those familiar in classical homology. Coefficients have a beautiful geometrical interpretation and the universal coefficient sequence is absorbed into the more general ‘killing’ exact sequence. Equivariant cohomology is easy to set up and operations are defined in a general setting. Finally there is the new concept of a generalised cohomology with a sheaf of coefficients (which unfortunately does not have all the nicest properties).”

These notes will provide matter for thought for all algebraic topologists, particularly those who were brought up to regard PL-theories as being difficult (since there is no Lie group corresponding to PL-manifolds in the same way that, e.g. \(O(n)\) corresponds to smooth unoriented manifolds.)

After a reasonable amount of preparation, the authors introduce mock bundles. A mock bundle over a ball complex \(K\) consists of a PL-projection \(E\overset {p}\rightarrow\vert K\vert\) such that for each \(\sigma\in K\), \(p^{-1}(\sigma)\) is a compact PL manifold of dimension \(q+\dim(\sigma)\) with boundary \(p^{-1}(\dot\sigma)\). Think of these as bundles (for functoriality) but impose a cobordism equivalence relation. One obtains the PL-cobordism theory.

Generalizations of this basic idea together with a related study of singularities of manifolds form the core of these notes.

The reviewer believes that the authors’ evident enthusiasm for their approach is justified. Particularly striking is the treatment of the transfer, the discussion of \(Z_2\) operations on PL-cobordism, and the fact that any CW spectrum may be viewed as the Thom spectrum of a suitable bordism theory.

The authors dedicate their work to Dennis Sullivan – indeed, these notes will serve to propagate his ideas.

The notes are not self-contained, but the authors carefully provide references for missing proofs. Some familiarity with PL-topology seems essential. Sophisticated ideas are treated on occasion in a deceptively simple manner. The notes are physically attractive with many line illustrations. They might be suitable for a graduate level seminar or for the mathematician who is simply curious to see how geometry is currently invading topology.

Reviewer: Claude Schochet

##### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |

55S25 | \(K\)-theory operations and generalized cohomology operations in algebraic topology |

55N30 | Sheaf cohomology in algebraic topology |

57R15 | Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) |

57Q20 | Cobordism in PL-topology |

57R90 | Other types of cobordism |

57Q99 | PL-topology |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

18G99 | Homological algebra in category theory, derived categories and functors |

57Q05 | General topology of complexes |