Vladimir V. Marchenko
Russian University of Peoples' Friendship, Moscow, Russia
Topological vector spaces of cochains and chains for Eucledian compacta
For an arbitrary compact $K$ of the Eucledian space $\R^n$ we
define the spaces $C^p(K)$ of germs of $p$-forms and $C_p(K)$ of $p$-currents on $K$ using the inverse system $\{U_\alpha,i_{\alpha\beta},A\}$ of all open neighborhoods $U_\alpha$ of $K$ in $\R^n$ and embeddings $i_{\alpha\beta}\colon U_\alpha \subseteq U_\beta$, $\alpha, \beta\in A$.
We raise the problem of studying $\R$-shape type of $K$. It turns out to be complicated enough. However, we obtain the following results.
Theorem 1. $C^p(K)$ and $C_p(K)$ are locally convex topological vector spaces. Moreover, $C^p(K)$ is also a commutative cochain space with an operation $\wedge$ extended from $C^p(U_\alpha)$ $(U_\alpha$ is open$)$ to $C^p(K), p=0, 1, \ldots$.
The spaces $H^p(K)$ and $H_p(K)$ of the de Rham cohomologies and homologies
for $K$ are defined as well.
Theorem 2. The spaces of the Chech cohomologies $\check H^p(K)$ and
Chech homologies $\check H_p(K)$ are isomorphic to the de Rham cohomologies
$H^p(K)$ and the de Rham homologies $H_p(K)$ of the compact $K$.
Unfortunately, $C^p(K)$ is not Hausdorff, but by slightly changing $C_p(K)$ we construct a new cochain space (we also denote it $C^p(K)$), which is Hausdorff, nuclear and barreled (but incomplete). The new $C_p(K)$ is nuclear and complete (but nonbarreled).
We construct new cochains $\tilde C^p(K)$ and chains $\tilde C_p(K)$ of the compact $K$, which are herediterily reflexive and dual to each other, and a continuous epimorphism (not topological homomorphim) $\varphi\colon\tilde C^p(K)\to C^p(K)$ and concider on $C^p(K)$ a new factor topology of $\left.\tilde C^p(K)\right/\ker\varphi$.
Theorem 3. In this new topology $C^p(K)$ becomes a reflexive topological vector space with the strong dual $C_p(K)$ (in new topology).
Cohomologies $H^p(K)$ and homologies $H_p(K)$ are reflexive and dual to each
other in the topologies induced from $C^p(K)$ and $C_p(K)$, respectively.
* This research was supported by the grant of Russia Fund of Fundamental Researches (Grant # 06-01-00341).
