3-manifold
Invariants describing 3-manifolds

3-manifolds are an interesting special case of low-dimensional topology because their topological invariants give a lot of information about their structure in general. If we let M {\displaystyle M} be a 3-manifold and π = π 1 (M) {\displaystyle \pi =\pi _{1}(M)} be its fundamental group, then a lot of information can be derived from them. For example, using Poincare duality and the Hurewicz theorem, we have the following homology groups:

H 0 (M) = H 3 (M) = Z H 1 (M) = H 2 (M) = π / H 2 (M) = H 1 (M) = Hom (π, Z) H 3 (M) = H 0 (M) = Z {\displaystyle {\begin{aligned}H_{0}(M)&=H^{3}(M)=&\mathbb {Z} \\H_{1}(M)&=H^{2}(M)=&\pi /\\H_{2}(M)&=H^{1}(M)=&{\text{Hom}}(\pi,\mathbb {Z})\\H_{3}(M)&=H^{0}(M)=&\mathbb {Z} \end{aligned}}}

where the last two groups are isomorphic to the group homology and cohomology of π {\displaystyle \pi }, respectively; that is,

H 1 (π; Z) ≅ π / H 1 (π; Z) ≅ Hom (π, Z) {\displaystyle {\begin{aligned}H_{1}(\pi ;\mathbb {Z})&\cong \pi /\\H^{1}(\pi ;\mathbb {Z})&\cong {\text{Hom}}(\pi,\mathbb {Z})\end{aligned}}}

From this information a basic homotopy theoretic classification of 3-manifolds can be found. Note from the Postnikov tower there is a canonical map

q: M → B π {\displaystyle q:M\to B\pi }

If we take the pushforward of the fundamental class ∈ H 3 (M) {\displaystyle \in H_{3}(M)} into H 3 (B π) {\displaystyle H_{3}(B\pi)} we get an element ζ M = q ∗ () {\displaystyle \zeta _{M}=q_{*}()}. It turns out the group π {\displaystyle \pi } together with the group homology class ζ M ∈ H 3 (π, Z) {\displaystyle \zeta _{M}\in H_{3}(\pi,\mathbb {Z})} gives a complete algebraic description of the homotopy type of M {\displaystyle M}.
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