Here is briefly the story. More details can be found in DeRham's monograph *Variétés differentiables*.

Let $M$ be a smooth, compact, oriented, $m$-dimensional manifold. Denote by $\Omega^k(M)$ the space of smooth degree $k$-forms on $M$ and by $\Omega_k(M)$ its dual space, namely the space of $k$-dimensional currents. Let $\newcommand{\bR}{\mathbb{R}}$

$$ \langle-,-\rangle :\Omega^k(M)\times \Omega_k(M)\to\bR,\;\; (\eta,C)\mapsto\langle \eta,C\rangle,
$$
denote the natural pairing between topological vector space and its dual.

We have a natural map

$$\cap [M]: \Omega^k(M)\to \Omega_{m-k},\;\;\Omega^k(M)\ni \alpha \mapsto \alpha\cap [M] \in \Omega_{m-k}(M), $$

determined by $\newcommand{\lan}{\langle}$ $\newcommand{\ran}{\rangle}$

$$\lan \eta, \alpha\cap [M]\ran :=\int_M \alpha\wedge \eta ,\;\;\forall \eta\in\Omega^{m-k}(M). $$

If we denote by $\newcommand{\pa}{\partial}$ $\pa:\Omega_k(M)\to\Omega_{k-1}(M)$ the boundary operator on $\Omega_\bullet(M)$ defined by
$$ \lan\alpha ,\pa C\ran:= \lan d\alpha, C\ran,\;\;\forall \alpha\in\Omega^{k-1}(M),\;\;C\in \Omega_k(M), $$

then we obtain a cochan complex

$$(\Omega_{m-\bullet}(M), \pa):\;\; \cdots \stackrel{\pa}{\to} \Omega_{m-k}(M)\stackrel{\pa}{\to}\Omega_{m-(k+1)}(M)\stackrel{\pa}\cdots . $$

We then have a morphism of cochain complexes

$$PD_M :(\Omega^\bullet(M), d) \to (\Omega_{m-\bullet}(M),\pa),\;\; \alpha \mapsto \alpha\cap [M]. $$

Georges DeRham proved that this morphism induces isomorphism in cohomology. This is one incarnation of Poincare duality. The cohomology of $(\Omega_{m-\bullet}(M),\pa)$ with the homology of $M$ with real coefficients. In your question you have identified $T_S$ with a $2r$-cohomology class using using $PD_M^{-1}$. This identification implicitly uses Poincarte duality.