# 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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*Poincaré duality complex of formal dimension*(

Tex syntax error-complex), , consists of an object in

Tex syntax errorwith finitely presented fundamental group , an orientation character, , viewed as a group homomorphism and a fundamental class

Tex syntax error, such that

Tex syntax error

*oriented*morphism of

Tex syntax error-complexes is a morphism in

Tex syntax error, such that and . The category

Tex syntax erroris the category consisting of

Tex syntax error-complexes and oriented or

*degree*morphisms of

Tex syntax error—complexes.

Let -

**types**be the full subcategory of the homotopy category

Tex syntax errorconsisting of

Tex syntax error-complexes in

Tex syntax errorwith for and let

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*Postnikov functor*. For , a

*fundamental triple*

*of formal dimension*consists of an -type , a homomorphism and an element

Tex syntax error. A morphism between fundamental triples is a homotopy class of maps of the -types, such that and . We obtain the category

Tex syntax errorof fundamental triples of formal dimension .

Every degree morphism in

Tex syntax errorinduces a surjection on fundamental groups, see for example [Browder1972a]. The category

Tex syntax erroris the subcategory consisting of all morphisms inducing surjections on fundamental groups, and we obtain the functor

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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

Tex syntax error, we first construct a chain map preserving fundamental classes, that is, . Then we use the category

Tex syntax errorof homotopy systems of order introduced in [Baues1991] to realize by a map with .

**Theorem 1.2.**Take . Two

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-connectedTex 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].

**Corollary 1.3.**For ,

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

Corollary 1.3 is Corollary 3.3 in [Baues&Bleile2008].

**Corollary 1.4.**Given a

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

Tex syntax errorand there is a surjection of groups

Tex syntax error

Corollary 1.4 is included in Corollary 4.4 in [Baues&Bleile2008].

**Remark 1.5.**For , let be the integer part of . Associating with a

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.

## 1 References

- [Baues&Bleile2008] H. J. Baues and B. Bleile,
*Poincaré duality complexes in dimension four*, Algebr. Geom. Topol.**8**(2008), no.4, 2355–2389. MR2465744 (2010b:57028) Zbl 1164.57008 - [Baues1991] H. J. Baues,
*Combinatorial homotopy and -dimensional complexes*, Walter de Gruyter & Co., 1991. MR1096295 (92h:55008) Zbl 0716.55001 - [Baues2003] H. Baues,
*The homotopy category of simply connected 4-manifolds*, Cambridge University Press, 2003. MR1996198 (2004g:57039) Zbl 1039.55009 - [Browder1972a] W. Browder,
*Poincaré spaces, their normal fibrations and surgery*, Invent. Math.**17**(1972), 191–202. MR0326743 (48 #5086) Zbl 0244.57007 - [Cavicchioli&Spaggiari2001] A. Cavicchioli and F. Spaggiari,
*On the homotopy type of Poincaré spaces*, Ann. Mat. Pura Appl. (4)**180**(2001), no.3, 331–358. MR1871619 (2002k:57053) Zbl 1034.57020 - [Hambleton&Kreck1988] I. Hambleton and M. Kreck,
*On the classification of topological -manifolds with finite fundamental group*, Math. Ann.**280**(1988), no.1, 85–104. MR928299 (89g:57020) Zbl 0616.57009 - [Hendriks1977] H. Hendriks,
*Obstruction theory in -dimensional topology: an extension theorem*, J. London Math. Soc. (2)**16**(1977), no.1, 160–164. MR0454980 (56 #13222) Zbl 03605632 - [Milnor1958] J. Milnor,
*On simply connected -manifolds*, Symposium internacional de topología algebraica International symposi um on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City (1958), 122–128. MR0103472 (21 #2240) Zbl 0105.17204 - [Swarup1974] G. A. Swarup,
*On a theorem of C. B. Thomas*, J. London Math. Soc. (2)**8**(1974), 13–21. MR0341474 (49 #6225) Zbl 0281.57003 - [Teichner1992] P. Teichner,
*Topological 4-manifolds with finite fundamental group*PhD Thesis, University of Mainz, Germany, Shaker Verlag 1992, ISBN 3-86111-182-9.

- [Thomas1969] C. B. Thomas,
*The oriented homotopy type of compact -manifolds*, Proc. London Math. Soc. (3)**19**(1969), 31–44. MR0248838 (40 #2088) Zbl 0167.21502 - [Turaev1989] V. G. Turaev,
*Three-dimensional Poincaré complexes: homotopy classification and splitting*, Mat. Sb.**180**(1989), no.6, 809–830, translation in Math. USSR-Sb. 67 (1990), 261–282. MR1015042 (91c:57031) Zbl 0717.57008 - [Wall1967a] C. T. C. Wall,
*Poincaré complexes. I*, Ann. of Math. (2)**86**(1967), 213–245. MR0217791 (36 #880) - [Whitehead1949] J. H. C. Whitehead,
*On simply connected, -dimensional polyhedra*, Comment. Math. Helv.**22**(1949), 48–92. MR0029171 (10,559d) Zbl 0039.39503

## 2 External links

- The Encyclopedia of Mathematics article about Poincaré complexes
- The Wikipedia page about Poincaré complexes
- Poincaré complex in n-Lab