A BESTIARY OF TOPOLOGICAL OBJECTS

 By N.P.Strickland here. I get a mention:

"12.3. Swarup’s homotopy classification. Let M be the category of 3-manifolds with a given basepoint. The morphisms from 𝑀 to 𝑀′ are homotopy classes of pointed maps which have degree one (in other words, we require that 𝑓∗[𝑀] = [𝑀′] ∈ 𝐻3𝑀′).

Let 𝒢 be the category of pairs (𝜋, 𝑢), where 𝜋 is a group and 𝑢 ∈ 𝐻3𝐵𝜋. The morphisms from (𝜋,𝑢)to(𝜋′,𝑢′)arehomomorphisms𝑓:𝜋→− 𝜋′ suchthat(𝐵𝑓)∗𝑢=𝑢′.

Given a manifold 𝑀 ∈ M, there is an obvious map 𝑞: 𝑀 →− 𝐵𝜋1𝑀, and we can define 𝜁𝑀 = 𝑞∗[𝑀] ∈ 𝐻3𝐵𝜋1𝑀. We thus get an object 𝐹𝑀 = (𝜋1𝑀,𝜁𝑀) of 𝒢, and it is easy to see that this gives a functor 𝐹 : M →− 𝒢 . Swarup proved that this functor is full [43]. It follows that 𝐹 𝑀 is isomorphic to 𝐹𝑀′ if and only if 𝑀 is homotopy equivalent to 𝑀′, by an orientation preserving equivalence. In other words, 𝐹𝑀 is a complete invariant of the homotopy type of 𝑀."