## Realization of homotopy invariants by *PD*^{3}-pairs

### Beatrice Bleile

#### Abstract

Up to oriented homotopy equivalence, a

*PD*^{3}-pair

*(X, ∂ X)* with
aspherical boundary components is uniquely determined by the

*Π*_{1}-system

*{ κ*_{i}:
Π_{1}(∂ X_{i}, *) →
Π_{1}(X, *) }_{i in J}, the
orientation character

*ω*_{X} in
H^{1}(X;**Z** / 2 **Z**) and the image of the fundamental
class

*[X, ∂ X] in H*_{3}(X, ∂ X;
**Z**^{ω}) under the classifying map. We call the
triple

*({ κ*_{i} }_{i in J},
ω_{X}, [X, ∂ X]) the fundamental triple
of the

*PD*^{3}-pair

*(X,∂ X)*.

Using
Peter Hilton's homotopy theory of modules, Turaev gave a
condition for realization in the absolute case of
*PD*^{3}-complexes *X* with *∂ X =
∅*. Given a finitely presentable group *G* and
*ω in H*^{1}(G;**Z** / 2 **Z**), he defined a
homomorphism * ν: H*_{3}(G; **Z**^{ω}) → [F,
I] where *F* is some **Z**[G]-module, *I = ker
aug* and *[A, B]* denotes the group of homotopy classes
of **Z**[G]-morphisms from the **Z**[G]-module *A*
to the **Z**[G]-module *B*. Turaev showed that, given
*µ in H*_{3}(G; **Z**^{ω}), the
triple *(G, ω, µ)* is relized by a
*PD*^{3}-complex *X* if and only if
*ν(µ)* is a class of homotopy equivalences of
**Z**[G]-modules.

Using Turaev's construction of the
homomorphism *ν*, we generalize the condition for
realization to the case of *PD*^{3}-pairs *(X,
∂ X)*, where *∂ X* is not necessarily
empty.

Keywords:
*PD*^{3}-Pairs.

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