Classifying Poincaré complexes via fundamental triples
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Let be the category of reduced CW-complexes, that is CW-complexes with -skeleton a point which is the base-point, and base-point preserving cellular maps. Given an object, , in , let be the universal cover of , and let be its cellular chain complex viewed as a complex of left modules over the integer group ring . To obtain a functor we assume that each object in is endowed with a base point in the universal covering over the base point of . Then a map in induces a unique base point preserving covering map which, in turn, induces a map ensuring that is functorial in .
Given a homomorphism , we define the anti-isomorphism of group rings, , by for and extending linearly to all of . Then, for a left -module, , the right module has the same underlying abelian group and action given by for and . For , we put
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Tex syntax error-complexes is a morphism in
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Tex syntax error-complexes and oriented or degree morphisms of
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Let -types be the full subcategory of the homotopy category
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Tex syntax error-complexes in
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Tex syntax error. A morphism between fundamental triples is a homotopy class of maps of the -types, such that and . We obtain the category
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Every degree morphism in
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Theorem 1.1. The functor reflects isomorphisms and is full for , that is, is surjective onto sets of morphisms and is an isomorphism if and only if is an isomorphism..
Tex syntax error-complexes and , , and a morphism in
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Tex syntax error-complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic.
Tex syntax error-complexes whose -type is an Eilenberg-Mac Lane space . Theorem 1.1 also yields a criterion for the existence of a map of degree one between
Tex syntax error-complexes, recovering Swarup's result for maps between -manifolds and Hendriks' result for maps between
Tex syntax error-complexes.
Special cases of Theorem 1.1 and 1.2 were proved by Hambleton and Kreck [Hambleton&Kreck1988] for . Teichner extended their approach to the non-oriented case in his thesis [Teichner1992]. Cavicchioli and Spaggiari [Cavicchioli&Spaggiari2001] studied the homotopy type of finite oriented Poincar\'e complexes in even dimensions.By early work of Milnor [Milnor1958] and Whitehead [Whitehead1949], the homotopy type of a simply-connected
Tex syntax error-complex, , is completely determined by its quadratic form. The -type of such an , with , is an Eilenberg-Mac Lane space
Tex syntax errorwith
Tex syntax error. The image of the fundamental class, , under the secondary boundary homomorphism in Whitehead's Certain Exact Sequence is the quadratic form of . Hence, in this case, the functor, , coincides with the functor
Tex syntax errorof Theorem 2.1.8 in [Baues2003].
Tex syntax error-complexes and and a map , there is a degree map, , rendering
homotopy commutative, if and only if induces a surjection on fundamental groups, is compatible with the orientations and , that is, , and
Tex syntax error-complex , let
Tex syntax errorbe the group of oriented homotopy equivalences of in
Tex syntax errorand
Tex syntax errorthe group of automorphisms of the triple in
Tex syntax error. Then the latter is a subgroup of
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Tex syntax error-complex, , the pre-fundamental triple , an orientation preserving map between
Tex syntax error-complexes is a homotopy equivalence if and only if the induced map between pre-fundamental triples is an isomorphism. However, pre-fundamental triples do not determine the homotopy type of a
Tex syntax error-complex, as is demonstrated by the fake products , where is the sum of the Whitehead product, , of the inclusions of the factors in the wedge product, and an element with a non-trivial element having trivial Hopf invariant. Pre-fundamental triples coincide with the fundamental triple for and . It remains an open problem to enrich the structure of a pre-fundamental triple to obtain an analogue of Theorem 1.2.
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