Generalized cohomology theories

extraordinary cohomology theories

A class of special functors from the category of pairs of spaces into the category of graded Abelian groups.

A generalized cohomology theory is a pair $ ( h ^ {*} , \delta ) $, where $ h ^ {*} $ is a functor from the category $ P $ of pairs of topological spaces into the category $ G A $ of graded Abelian groups (that is, to each pair of spaces $ ( X, A ) $ corresponds a graded Abelian group $ h ^ {*} ( X, A ) = \oplus _ {n = - \infty } ^ \infty h ^ {n} ( X , A ) $ and to each continuous mapping $ f : ( X, A ) \rightarrow ( Y , B ) $ a set of homomorphisms $ \\{ h ^ {n} ( f ) : h ^ {n} ( Y , B ) \rightarrow h ^ {n} ( X , A ) \\} _ {n = - \infty } ^ \infty $), and $ \delta $ is a set of homomorphisms

$$ \\{ \delta _ {( X , A ) } ^ {n} : h ^ {n} ( A) \rightarrow h ^ {n+} 1 ( X , A ) \\} , $$

given for each pair $ ( X , A ) $ and natural in the sense that for any continuous $ f : ( X , A ) \rightarrow ( Y , B ) $ the following equation holds:

$$ \delta _ {( X , A ) } ^ {n} \circ h ^ {n} ( f \mid _ {A} ) = h ^ {n} ( f ) \circ \delta _ {( Y , B ) } ^ {n} , $$

and the following three axioms must be satisfied.

1) The homotopy axiom. If two mappings $ f , g : ( X , A ) \rightarrow ( Y , B ) $ are homotopic, then the homomorphisms $ h ^ {n} ( f ) $ and $ h ^ {n} ( g) $ are the same for all $ n $.

2) The exactness axiom. For any pair $ ( X , A ) $ the sequence

$$ {} \dots \rightarrow h ^ {n} ( X , A ) \rightarrow ^ { {h ^ {n}} ( j) } \ h ^ {n} ( X) \rightarrow ^ { {h ^ {n}} ( i) } h ^ {n} ( A) \rightarrow ^ { {\delta _ {(} X , A ) } ^ {n} } $$

$$ \rightarrow ^ { {\delta _ {(} X , A ) } ^ {n} } h ^ {n+} 1 ( X , A ) \rightarrow ^ { {h ^ {n+}} 1 ( j) } \dots $$

is exact; here $ i : A \rightarrow X $ and $ j : X = ( X , \emptyset ) \rightarrow ( X , A ) $ are the obvious inclusions.

3) The excision axiom. Let $ ( X , A ) $ be a pair of spaces and let $ U \subset A $ be such that $ \overline{U}\; \subset A $. Then the inclusion $ i : ( X \setminus U , A \setminus U ) \rightarrow ( X , A ) $ induces, for all $ n $, isomorphisms

$$ h ^ {n} ( X , A ) \rightarrow h ^ {n} ( X \setminus U , A \setminus U ) . $$

For a cofibration $ ( X , A ) $ it follows from the axioms that the projection $ ( X , A ) \rightarrow ( X / A , \mathop{\rm pt} ) $, where $ \mathop{\rm pt} $ is a space consisting of a single point, induces an isomorphism

$$ h ^ {n} ( X , A ) \rightarrow h ^ {n} ( X \setminus U , A \setminus U ) . $$

Often one simply writes $ f ^ { * } $ instead of $ h ^ {n} ( f ) $ and $ \delta $ instead of $ \delta _ {( A , X ) } ^ {n} $. The group $ h ^ {n} ( X , A) $ is called the $ n $- dimensional (generalized) cohomology group of the pair $ ( X , A ) $, and the graded group $ h ^ {*} ( \mathop{\rm pt} ) $ is called the coefficient group of the generalized cohomology theory.

In the definition of a generalized cohomology theory the category $ P $ can be replaced by the category of pairs of cofibrations or by the category $ \mathfrak S $ of pairs of CW-complexes or by the category $ \mathfrak S _ {F} $ of pairs of finite CW-complexes (here, in the excision axiom one must require that the pair $ ( X \setminus U , A \setminus U ) $ is isomorphic to an object in the appropriate category). In these cases one says that the generalized cohomology theory $ ( h , \delta ) $ is defined on the category $ \mathfrak S $( respectively, $ \mathfrak S _ {F} $).

The choice of the term "generalized cohomology theory" is justified by the following circumstances. It was proved in [2] that any functor $ \mathfrak S _ {F} \rightarrow G A $ satisfying axioms 1)–3) and the so-called dimension axiom (which states that $ h ^ {i} ( \mathop{\rm pt} ) = 0 $ for $ i \neq 0 $) is the usual cohomology theory $ H ^ {*} $ with coefficients in $ h ^ {0} ( \mathop{\rm pt} ) $. Later it was noticed that many useful constructions in algebraic topology (for example, cobordism;  $ K $- theory) satisfy axioms 1)–3) and that the effectivity of these constructions depends to a significant extent on properties which follow formally from these axioms. This led to the acceptance of the concept of generalized cohomology theories, which had been formulated earlier.

Let $ X $ be a pointed space and let $ \epsilon : \mathop{\rm pt} \rightarrow X $ be its basepoint. The reduced generalized cohomology group $ \widetilde{h} {} ^ {n} ( X) $ of $ X $ is defined by putting

$$ \widetilde{h} {} ^ {n} ( X) = \ \mathop{\rm ker} ( h ^ {n} ( \epsilon ) \cdot h ^ {n} ( X) \rightarrow h ^ {n} ( \mathop{\rm pt} ) ) . $$

There is an obvious splitting

$$ h ^ {n} ( X) = \widetilde{h} {} ^ {n} ( X) \oplus h ^ {n} ( \mathop{\rm pt} ) , $$

and this splitting is canonical, noting that the inclusion $ h ^ {n} ( \mathop{\rm pt} ) \subset h ^ {n} ( X) $ is induced by the mapping $ X \rightarrow \mathop{\rm pt} $. It is clear that $ \widetilde{h} {} ^ {n} ( X) \approx h ^ {n} ( X , \mathop{\rm pt} ) $. Also, it follows from 1)–3) that for a cofibration $ ( X , A ) $ there is an isomorphism $ h ^ {n} ( X , A ) \approx h ^ {n} ( X / A , \mathop{\rm pt} ) $( see [2], [3]), so that $ h ^ {n} ( X , A ) \approx \widetilde{h} {} ^ {n} ( X / A ) $. Here, as usual, $ X / A = X \cup \mathop{\rm pt} = X ^ {t} $ for $ A = \emptyset $.

If $ ( X , A ) $ is a cofibration, then it follows from the axioms that the sequence

$$ \tag{* } {} \dots \rightarrow \widetilde{h} {} ^ {n} ( X / A ) \rightarrow ^ { {j ^ {*}} } \ \widetilde{h} {} ^ {n} ( X) \rightarrow ^ { {i ^ {*}} } \ \widetilde{h} {} ^ {n} ( A) \mathop \rightarrow \limits ^ \delta $$

$$ \mathop \rightarrow \limits ^ \delta \widetilde{h} {} ^ {n+} 1 ( X / A ) \rightarrow \dots $$

is exact (it is natural in the category of cofibrations). Here $ i : A \rightarrow X $ and $ j : X \rightarrow X / A $ are the obvious mappings and $ \delta $ is the composition

$$ \widetilde{h} {} ^ {n} ( A) \subset h ^ {n} ( A) \rightarrow \ h ^ {n+} 1 ( X , A ) \approx \ \widetilde{h} {} ^ {n+} 1 ( X / A ) . $$

In particular, if $ X $ is the cone $ C A $ on $ A $( cf. Mapping-cone construction), then $ \widetilde{h} ( X) = 0 $( the homotopy axiom), and $ X / A $ is the suspension $ S A $ of $ A $; the exactness of the sequence (*) implies that there is a suspension isomomorphism $ \sigma _ {A} : \widetilde{h} {} ^ {i} ( A) \rightarrow \widetilde{h} {} ^ {i+} 1 ( S A ) $, natural with respect to $ A $. Here, the isomorphism $ \sigma $ allows one to reconstruct $ \delta $( see [2], [3]); this is done by means of the so-called Puppe sequence. Applying the functor $ h ^ {N} $, as $ N \rightarrow \infty $, to the latter sequence gives the exactness of (*). Thus, the generalized cohomology theory $ ( h ^ {*} , \delta ) $ can be completely reconstructed in terms of the reduced theory $ ( \widetilde{h} {} ^ {*} , \sigma ) $.

A generalized cohomology theory $ h ^ {*} $ is called multiplicative if for any pairs of spaces $ ( X , A ) $, $ ( Y , B ) $ in $ P $ there is given a natural pairing

$$ h ^ {p} ( X , A ) \oplus h ^ {q} ( Y , B ) \rightarrow h ^ {p+} q ( X \times Y ,\ X \times B \cup A \times Y ) $$

satisfying the conditions of graded commutativity and associativity (see [4], [5]). In this case, for $ ( X , A ) \in P $, the group $ h ^ {*} ( X , A) $ is a graded (commutative, associative) ring with respect to the multiplication

$$ h ^ {p} ( X , A ) \oplus h ^ {q} ( X , A ) \rightarrow h ^ {p+} q ( X \times X , X \times A \cup A \times X ) \ \rightarrow ^ { {\Delta ^ {*}} } \ $$

$$ \rightarrow ^ { {\Delta ^ {*}} } h ^ {p+} q ( X , A ) , $$

where

$$ \Delta : ( X , A ) \rightarrow ( X \times X ,\ A \times A ) \subset ( X \times X , X \times A \cup A \times X ) $$

is the diagonal mapping, and the induced mappings $ f ^ { * } : h ^ {*} ( Y, B ) \rightarrow h ^ {*} ( X , A ) $ are ring homomorphisms. More generally, pairings of two generalized cohomology theories into a third may be defined [5].

The ordinary cohomology $ H ^ {n} ( X ; G ) $ can be defined as the group $ [ X , K ( G , n ) ] $ of homotopy classes of continuous mappings of $ X $ into the Eilenberg–MacLane space $ K ( G , n ) $. This can be extended to generalized cohomology theories as follows. A sequence of spaces $ \\{ M _ {n} \\} _ {n= - \infty } ^ \infty $ and continuous mappings $ s _ {n} : S M _ {n} \rightarrow M _ {n+} 1 $, where $ S M _ {n} $ is the suspension of $ M _ {n} $, is called a spectrum of spaces. For a space $ X $ the group $ \widetilde{h} {} ^ {n} ( X) $ is defined by the equation

$$ \widetilde{h} {} ^ {n} ( X) = \ \lim\limits _ {k \rightarrow \infty } \ [ S ^ {k} X , M _ {n+} k ] . $$

Here, the mapping

$$ [ S ^ {k} X , M _ {n+} k ] \rightarrow \ [ S ^ {k+} 1 X , M _ {n+} k+ 1 ] $$

is defined as the composition

$$ [ S ^ {k} X , M _ {n+} k ] \rightarrow ^ { S } \ [ S ^ {k+} 1 X , S M _ {n+} k ] \ \mathop \rightarrow \limits ^ { {( s _ {n+} k ) }} \ [ S ^ {k+} 1 X , M _ {n+} k+ 1 ] . $$

The suspension isomorphisms $ \sigma _ {X} ^ {n} : \widetilde{h} {} ^ {n} ( X) \rightarrow \widetilde{h} {} ^ {n+} 1 ( S X ) $ are constructed in the obvious way. Thus, each spectrum of spaces gives a certain generalized cohomology theory $ ( \widetilde{h} {} ^ {*} , \sigma ) $ and, hence, an unreduced generalized cohomology theory $ ( h ^ {*} , \delta ) $.

If, given a generalized cohomology theory $ ( h ^ {*} , \delta ) $, there exists a spectrum from which it is obtained by the above method, then one says that this spectrum represents $ ( h ^ {*} , \delta ) $, or that the theory $ ( h ^ {*} , \delta ) $ is representable by this spectrum. It is known that any generalized cohomology theory on the category $ \mathfrak S _ {F} $ is representable by a spectrum [5].

If $ ( h ^ {*} , \delta ) $ is representable by a ringed spectrum of spaces, then it is multiplicative [5]. For a generalized cohomology theory given on the category $ \mathfrak S $ the converse is also true.

Let $ F \rightarrow E \rightarrow B $ be a Serre fibration. For any generalized cohomology theory $ h ^ {*} $ and any $ n $, the groups $ h ^ {n} ( F ) $ form a local system of groups on $ B $. There exists the Dold–Atiyah–Hirzebruch spectral sequence $ \\{ E _ {r} ^ {p,q} \\} $, with initial term $ E _ {2} ^ {p,q} = H ^ {p} ( B ; \\{ h ^ {q} ( F ) \\} ) $. If $ B $ is a finite CW-complex, then this spectral sequence converges and its limit term is associated to $ h ^ {*} ( E) $( see [1]). In particular, if $ F = \mathop{\rm pt} $, then one obtains the spectral sequence $ H ^ {p} ( X , h ^ {q} ( \mathop{\rm pt} ) ) \Rightarrow h ^ {n} ( X) $, (sometimes) allowing the group $ h ^ {*} ( X) $ to be computed in terms of $ H ^ {*} ( X) $ and $ h ^ {*} ( \mathop{\rm pt} ) $.

With each generalized cohomology theory $ h ^ {*} $ one can associate a dual generalized homology theory $ h _ {*} $, whose axioms are analogous to those for a generalized cohomology theory except that homology is a covariant functor [4]. Here, if the spaces $ X $ and $ Y $ are $ ( n + 1 ) $- dual (see  $ S $- duality) then $ \widetilde{h} {} ^ {i} ( X) \approx \widetilde{h} _ {n-} i ( Y) $. Also, if $ h ^ {*} $ is representable by the spectrum $ \\{ M _ {n} , s _ {n} \\} $, then

$$ \widetilde{h} _ {i} ( X) = \ \lim\limits _ {k \rightarrow \infty } \ \pi _ {i+} k ( X \wedge M _ {k} ) . $$

Here, for a multiplicative generalized cohomology theory $ h ^ {*} $ there is an intersection pairing $ \cap $:

$$ \cap : h _ {n} ( X , X _ {1} \cup X _ {2} ) \oplus h ^ {q} ( X , X _ {1} ) \rightarrow h _ {n-} q ( X , X _ {2} ) . $$

The most important examples of generalized cohomology theories are  $ K $- theory and the various cobordism theories. The generalized homology theories dual to cobordisms are the bordisms (cf. Bordism).

Let $ \xi $ be an $ n $- dimensional vector bundle over $ X $, orientable (see Orientation) in a generalized cohomology theory $ h ^ {*} $, and let $ T \xi $ be its Thom space. In this case the generalized Thom isomorphism $ h ^ {i} ( X) \approx \widetilde{h} {} ^ {i+} n ( T \xi ) $ holds (see [1]). From this (and the Milnor–Spanier–Atiyah duality theorem [7]) follows the generalized Poincaré duality: Let $ P $ be a Poincaré space of formal dimension $ n $( for example, a closed $ n $- dimensional manifold) whose normal bundle is orientable in $ h ^ {*} $. Then for any integer $ i $ one has $ h _ {i} ( P) \approx h ^ {n-} i ( P) $. Let $ \nu $ be the $ N $- dimensional normal bundle over $ P $ and let $ T \nu $ be its Thom space. The spaces $ P ^ {+} = P \cup \mathop{\rm pt} $ and $ T \nu $ are $ ( N + n ) $- dual (the relation called $ ( n + 1 ) $- duality in the article  $ S $- duality is often called $ n $- duality). Therefore

$$ h _ {i} ( P) \approx \widetilde{h} _ {i} ( P ^ {+} ) \approx \widetilde{h} {} ^ {N+} n- i ( T \nu ) \approx h ^ {n-} i ( P) . $$

The element $ z $ in $ h _ {n} ( P) $ corresponding to the identity $ 1 \in h ^ {0} ( P) $ under this isomorphism is called the fundamental class of $ P $ in the theory $ h ^ {*} $; this generalizes the classical concept of a fundamental class. It can be shown that the isomorphism $ h ^ {i} ( P) \approx h _ {n-} i ( P) $ is given by "intersection with the fundamental class" , that is, it has the form $ x \rightarrow z \cap x $( see [4]).

Let $ F $ be one of the fields $ \mathbf R $ or $ \mathbf C $, or the skew-field of quaternions $ \mathbf H $. A multiplicative generalized cohomology theory $ h ^ {*} $ is called $ F $- orientable if all $ F $- vector bundles are orientable in $ h ^ {*} $. It turns out that for any $ F $- orientable theory $ h ^ {*} $ and any $ F $- vector bundle over $ X $ one can define the generalized characteristic classes (cf. Characteristic class) of a fibration $ \xi $ with values in the group $ h ^ {*} ( X) $; here, if $ F $ is equal to $ \mathbf R $, $ \mathbf C $ or $ \mathbf H $, and if one uses the ordinary cohomology theory $ H ^ {*} $( or $ H ^ {*} ( \mathbf Z _ {2} ) $ for $ F = \mathbf R $), then one obtains the Stiefel, Chern or Borel classes, respectively. In this context the theory of $ G F $- cobordism (see Cobordism) is a universal $ F $- orientable generalized cohomology theory. This is also clear from the existence of the spectral sequence connecting $ h ^ {*} ( X) $ with $ G F _ {0} ^ {*} ( X) $ and $ h ^ {*} ( \mathop{\rm pt} ) $, where $ F _ {0} $ is either $ \mathbf R $ or $ \mathbf C $. In addition, a formal group over the ring $ h ^ {*} ( \mathop{\rm pt} ) $ can be associated with each $ \mathbf C $- orientable generalized cohomology theory $ h ^ {*} $, and the universality of cobordisms is reflected in the fact that the formal group of the theory of unitary cobordism is universal (purely algebraically) in the class of all formal groups. Moreover, the formal group of the theory $ h ^ {*} $ carries quite a lot of information on $ h ^ {*} $.

It often becomes necessary to extend a generalized cohomology theory from a subcategory to the whole category. For example, it may be necessary to extend a theory $ h ^ {*} $ given on the category $ \mathfrak S _ {F} $ to the whole category $ \mathfrak S $.

First method: A spectrum representing $ h ^ {*} $( on $ \mathfrak S _ {F} $) is chosen, and by means of it the theory is extended to the whole of $ \mathfrak S $.

Second method: Let the theory $ h ^ {*} $ be given on $ \mathfrak S _ {F} $ and let $ X \in \mathfrak S $; suppose that $ \\{ X _ \alpha \\} $ is an exhausting family of finite CW-subspaces of $ X $ and set

$$ h ^ \leftarrow ( X) = \ {\lim\limits _ \leftarrow } {h ^ {*} ( X _ \alpha ) } . $$

Then $ {h ^ {*} } ^ \leftarrow $ is a functor on $ \mathfrak S $ satisfying all the axioms for a generalized cohomology theory except the exactness axiom (the functor $ \lim\limits _ \leftarrow $ does not preserve exactness). Thus, for any $ X \in \mathfrak S $ and any generalized cohomology theory $ h ^ {*} : \mathfrak S \rightarrow G A $ extending $ h ^ {*} : \mathfrak S _ {F} \rightarrow G A $, the natural homomorphism

$$ h ^ {*} ( X ) \rightarrow {h ^ {*} } ^ \leftarrow ( X) $$

is epimorphic.

In the general case the spectral sequence $ E _ {r} ^ {**} ( X) \Rightarrow h ^ {*} ( X) $ appears, where $ E _ {r} ^ {p,q} ( X) = \lim\limits ^ {p} \\{ h ^ {q} ( X _ \alpha ) \\} $; here the $ \lim\limits _ \rightarrow ^ {p} $ are the higher derived functions of $ \lim\limits _ \rightarrow $, see [10].

For a generalized homology theory $ h _ {*} $, given on $ \mathfrak S _ {F} $, the functor

$$ \vec{h} _ {*} ( X) = \ {\lim\limits _ \rightarrow } h _ {*} ( X _ \alpha ) $$

satisfies the exactness axiom, and hence is always an extension of $ h _ {*} $ from $ \mathfrak S _ {F} $ to $ \mathfrak S $.

The third method is an analogue of the Aleksandrov–Čech method and depends on using the construction of a nerve (cf. Nerve of a family of sets).

Generalized cohomology theories can also be extended to the category of spectra. Let $ M = \\{ M _ {n} , S _ {n} \\} $ be a spectrum of spaces. The group $ \widetilde{h} {} ^ {*} ( M) $ is defined by the relation

$$ \widetilde{h} {} ^ {i} ( M) = \ \lim\limits _ {n \rightarrow \infty } \ \widetilde{h} {} ^ {i+} n ( M _ {n} ) , $$

and the mappings

$$ \widetilde{h} {} ^ {i+} n ( M _ {n} ) \leftarrow \ \widetilde{h} {} ^ {i+} n+ 1 ( M _ {n+} 1 ) $$

have the form

$$ \widetilde{h} {} ^ {i+} n ( M _ {n} ) \approx \ \widetilde{h} {} ^ {i+} n+ 1 ( S M _ {n} ) \leftarrow \ \widetilde{h} {} ^ {i+} n+ 1 ( M _ {n+} 1 ) . $$

The resulting functor $ h ^ {*} $ on the category of spectra satisfies all the axioms for a reduced generalized cohomology theory (when transferred properly to the category of spectra) (see [5]).

There is a natural problem of "comparing" different generalized cohomology theories, and, in particular, the problem of expressing one cohomology theory in terms of another. The solution of the latter problem can be regarded as a far-reaching generalization of the universal coefficients formula. Spectral sequences of Adams type are the most powerful tool here. One such example has already been mentioned: The spectral sequences "from cobordisms to oriented generalized cohomology theories" . Another example: Let $ h ^ {*} $ and $ k ^ {*} $ be two generalized cohomology theories. Assume further that $ A _ {n} $ is the ring of cohomology operations (cf. Cohomology operation) of $ h ^ {*} $, that $ Y $ is a spectrum representing $ k ^ {*} $, and that $ X $ is some spectrum (in particular, a space). Then (for "good X, Y and h*" , see [6]) there exists a spectral sequence with initial term $ \mathop{\rm Ext} _ {A _ {h} } ^ {**} ( h ^ {*} ( Y) , h ^ {*} ( X) ) $, and with limit term conjugate with $ k ^ {*} ( X) $. There are also other spectral sequences (see [8], [9]) connecting one generalized cohomology theory with others.

It would be useful to learn how to treat a generalized cohomology theory as a cohomology functor, that is, to split $ h ^ {*} : P \rightarrow G A $ into the composition $ P \rightarrow ^ {i} A \rightarrow G A $, where $ i $ is a canonical functor (not depending on $ h ^ {*} $) into an Abelian category $ A $. One way of realizing this is outlined in [8].

#### References[edit]

[1] |  A. Dold, "Relations between ordinary and extraordinary homology" , Colloq. Algebraic Topology, August 1–10, 1962 , Inst. Math. Aarhus Univ. (1962) pp. 2–9 Zbl 0145.20104

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[2] |  S. Eilenberg, N.E. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1952) MR0050886 Zbl 0047.41402
[3] |  P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964) MR0176478 Zbl 0125.40103
[4] |  G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) MR0309097 Zbl 0217.48601
[5] |  R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975) MR0385836 Zbl 0305.55001
[6] |  S.P. Novikov, "The method of algebraic topology from the viewpoint of cobordism theories" Math. USSR-Izv. , 1 (1967) pp. 827–913 Izv. Akad. Nauk SSSR Ser. Mat. , 31 : 4 (1967) pp. 855–951
[7] |  M.F. Atiyah, "Thom complexes" Proc. London Math. Soc. , 11 (1961) pp. 291–310 MR0131880 Zbl 0124.16301
[8] |  J.F. Adams, "Stable homotopy and generalised homology" , Univ. Chicago Press (1974) MR0402720 Zbl 0309.55016
[9] |  E. Dyer, D. Kahn, "Some spectral sequences associated with fibrations" Trans. Amer. Math. Soc. , 145 (1969) pp. 397–437 MR0254840 Zbl 0191.53906
[10] |  S. Araki, Z. Yosimura, "A spectral sequence associated with a cohomology theory of infinite CW-complexes" Osaka J. Math. , 9 : 3 (1972) pp. 351–365 MR0326733 Zbl 0253.55013
[11] |  E. Dyer, "Cohomology theories" , Benjamin (1969) MR0268883 Zbl 0182.57002

#### Comments[edit]

For the Puppe sequence cf. the third part of the article Cone.