The uniqueness theorem for cohomologies on the category of polyhedral pairs

Let $X$ be a topological space and $F=\left\{F_{\alpha }\right\}$ be a direct system of all compact subsets $F_{\alpha }$ of $X$, directed by inclusions. For any homology theory $H_{\ast }$ the groups $\{H_{\ast }\left(F_{\alpha }\right)\mid F_{\alpha }\subset X\}$ constitute a direct system, and the maps $H_{\ast }\left(F_{\alpha }\right)\to H_{\ast }\left(X\right)$ define a homomorphism $i_{\ast }:\underset{⟶}{lim}{H}_{\ast }\left(F_{\alpha }\right)\to H_{\ast }\left(X\right)$.

As is known (Theorem 4.4.6, Spanier, 1966), for the singular homology, the homomorphism $i_{\ast }$ is an isomorphism (1)$i_{\ast }:\underset{⟶}{lim}{H}_{\ast }^s\left(F_{\alpha }\right)\stackrel{\sim }{⟶}{H}_{\ast }^s\left(X\right).$Using the isomorphism (1), it is proved that for the homologies having compact support $H$ there is the uniqueness theorem on the category of polyhedral pairs (Theorem 4.8.14, Spanier, 1966).

Since the singular homology theory is a homology theory with compact supports, the uniqueness theorem connects all homology theories having compact supports with the singular homology theory.

Let $H^{\ast }$ be a cohomology theory. The groups $\{H^{\ast }\left(F_{\alpha }\right)\mid F_{\alpha }\subset X\}$ constitute an inverse system, and the maps $H^{\ast }\left(X\right)\to H^{\ast }\left(F_{\alpha }\right)$ define a homomorphism $i^{\ast }:H^{\ast }\left(X\right)\to \underset{⟵}{lim}{H}^{\ast }\left(F_{\alpha }\right).$Since the homology functor does not commute with inverse limits, it is not true that the singular cohomology of a space is isomorphic to the inverse limit of the singular cohomology of its compact subsets (that is, there is no general cohomology analogue of Theorem 4.4.6, Spanier, 1966).

In the present work, it will be shown that there is such connection for a singular cohomology. Namely, there exists a finite exact sequence (2)$0⟶{\underset{⟵}{lim}}^{(2n-3)}{H}_s^1(F_{\alpha },G)⟶\cdots ⟶{\underset{⟵}{lim}}^{\left(1\right)}{H}_s^{n-1}(F_{\alpha },G)⟶H_s^n(X,G)⟶\underset{⟵}{lim}{H}_s^n(F_{\alpha },G)⟶{\underset{⟵}{lim}}^{\left(2\right)}{H}_s^{n-1}(F_{\alpha },G)⟶\cdots ⟶{\underset{⟵}{lim}}^{(2n-2)}{H}_s^1(F_{\alpha },G)⟶0.$

The terms the Alexander cohomology with compact supports and the singular cohomology with compact supports used in the works (Spanier, 1966; Mdzinarishvili, 1984) do not refer to our problem. Therefore, cohomology theory, in particular the singular cohomology, for which there is a finite exact sequence (2), is called a cohomology with partially compact supports.

In the present work, using a finite exact sequence (2), it is proved the uniqueness theorem for a cohomology having partially compact supports on the category of polyhedral pairs. Hence, the uniqueness theorem connects all cohomology theories with partially compact supports with the singular cohomology theory.