Group with Poincaré duality

Groups with Poincaré duality (English: Poincaré duality groups ) are a term from the mathematical field of group theory that is important in numerous questions of algebraic and geometric topology .

An open conjecture by Wall says that a finitely presented group fulfills a -dimensional Poincaré duality if and only if it is the fundamental group of an aspheric closed manifold . ${\displaystyle n}$

Definitions

A group is a group with -dimensional Poincaré duality if there is a as a module to isomorphic module is, one that for each module over the group ring and all one isomorphism ${\displaystyle G}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} G}$ ${\displaystyle D}$ ${\displaystyle A}$ ${\displaystyle \mathbb {Z} G}$ ${\displaystyle 0\leq i\leq n}$

${\displaystyle H^{i}(G,A)\simeq H_{n-i}(G,D\otimes A)}$

of th group cohomology with coefficients in the th group homology with coefficients in has. ${\displaystyle i}$ ${\displaystyle A}$ ${\displaystyle (n-i)}$ ${\displaystyle D\otimes A}$

For finitely presented groups this definition is equivalent to the condition that

• ${\displaystyle H^{i}(G,\mathbb {Z} G)=0}$for everyone and ${\displaystyle i\not =n}$
• ${\displaystyle H^{n}(G,\mathbb {Z} G)=\mathbb {Z} }$ applies.

Also for finitely presented groups indicates an equivalent definition is that a group met -dimensional Poincaré duality if they freely and properly discontinuously on a collapsible cell complex with (for cohomology with compact support acts). ${\displaystyle n}$ ${\displaystyle X}$ ${\displaystyle H_{c}^{*}(X,\mathbb {Z} )=H_{c}^{*}(\mathbb {R} ^{n},\mathbb {Z} )}$

Examples

The fundamental group of an aspherical closed -dimensional manifold fulfills -dimensional Poincaré duality. In fact, in this case the homologs and cohomology of the group are isomorphic to the homology and cohomology of the manifold, and Poincaré duality applies to the latter . The module is in this case the orientation module , which in the case of orientable manifolds with trivial is -Wirkung. ${\displaystyle G=\pi _{1}M}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle \mathbb {Z} G}$ ${\displaystyle D}$ ${\displaystyle D=H^{n}(G,\mathbb {Z} G)}$ ${\displaystyle D=\mathbb {Z} }$ ${\displaystyle G}$

The fundamental groups of closed manifolds are always finite presented. But there are also groups with Poincaré duality that are not finite presented.

properties

Be a group with -dimensional Poincaré duality. Then applies ${\displaystyle G}$ ${\displaystyle n}$

• ${\displaystyle G}$is finitely generated
• the cohomological dimension of is ${\displaystyle G}$ ${\displaystyle n}$
• ${\displaystyle G}$is torsion-free
• a subgroup is just then a group -dimensional Poincaré duality if finite index in has ${\displaystyle H\subset G}$ ${\displaystyle n}$ ${\displaystyle G}$
• a subgroup is a group with -dimensional Poincaré duality if and only if its cohomological dimension is ${\displaystyle H\subset G}$ ${\displaystyle n}$ ${\displaystyle n}$
• Subsets of infinite index have a cohomological dimension less than . ${\displaystyle n}$

literature

• KS Brown, Cohomology of Groups , Springer-Verlag, New York (1982).

Web links

• M. Davis: Poincaré duality groups

Individual evidence

1. ↑ M. Bestvina, N. Brady, Morse theory and finiteness properties of groups , Invent. Math. 129: 445-470 (1997).