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
Tex syntax error

Tex syntax error-complex),
![X = (X, \omega_X, [X])](/images/math/2/4/1/241cc76ef5fd0d69aec4b7fe26f19271.png)

Tex syntax errorwith finitely presented fundamental group



Tex syntax error, such that
Tex syntax error

![\mathbb Z [\pi_1 X]](/images/math/8/b/6/8b6e4a443a8f7fb3891c745cf136cd4b.png)

Tex syntax error-complexes
![f: (X, \omega_X, [X]) \rightarrow (Y, \omega_Y, [Y])](/images/math/2/5/5/255a1c45683f0c1f2e82c6132c9268de.png)

Tex syntax error, such that

![f_{\ast}[X] = [Y]](/images/math/b/e/a/bea4c05d0f3c489adcd89e9eefb4fc50.png)
Tex syntax erroris the category consisting of
Tex syntax error-complexes and oriented or degree

Tex syntax error—complexes.
Let

Tex syntax errorconsisting of
Tex syntax error-complexes

Tex syntax errorwith


Tex syntax error







Tex syntax error. A morphism





Tex syntax errorof fundamental triples


Every degree


Tex syntax errorinduces a surjection

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




Tex syntax error, we first construct a chain map

![\xi_{\ast}[Y] = [X]](/images/math/c/6/4/c6448e8a4adcf8e23031a017eb83065b.png)
Tex syntax errorof homotopy systems of order





Tex syntax error-complexes are orientedly homotopy equivalent if and only if their fundamental triples are isomorphic.

Tex syntax error-complexes whose


Tex syntax error-complexes, recovering Swarup's 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.
Tex syntax error-complex,




Tex syntax errorwith
Tex syntax error. The image of the fundamental class,
![[X]](/images/math/b/6/9/b697f334e90f4eaaec7edab7dc1a384c.png)


Tex syntax errorof Theorem 2.1.8 in [Baues2003].

Tex syntax error-complexes





![\displaystyle \xymatrix{ X \ar[r]^-{p_{n-2}} \ar@{..>}[d]_{\overline f} & P_{n-2}X \ar[d]^f \\ Y \ar[r]^-{p_{n-2}} & P_{n-2}Y}](/images/math/9/8/c/98c15ca585831a4015f3e759ad7312a5.png)
homotopy commutative, if and only if induces a surjection on fundamental groups, is compatible with the orientations
and
, that is,
, and
![\displaystyle f_{\ast}p_{n-2 \ast}[X] = p_{n-2 \ast}[Y].](/images/math/9/7/7/97700d99e1f786afe00d3335c4f082d7.png)
Corollary 1.3 is Corollary 3.3 in [Baues&Bleile2008].
Tex syntax error-complex

Tex syntax errorbe the group of oriented homotopy equivalences of

Tex syntax errorand
Tex syntax errorthe group of automorphisms of the triple

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].

![[\frac{n}{2}]](/images/math/c/e/3/ce35bed745e577a468dbf05fc57b225e.png)

Tex syntax error-complex,

![(P_{[\frac{n}{2}]}X, \omega_X, p_{[\frac{n}{2}]\ast}[X])](/images/math/d/2/6/d2646516bef77dcdf2d18fbdf672add9.png)
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


![[\iota_1 , \iota_2]](/images/math/a/c/d/acd39e7ffdf87edd607e66cb5ca4d891.png)




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