De Rham currents of compact oriented source cycles

Let $X$ be a compact oriented smooth manifold of dimension $n$ without boundary. For $0\le r\le n$, write ${\mathrm{\Omega }}^r(X)$ for the real smooth differential $r$-forms. Give ${\mathrm{\Omega }}^r(X)$ its $C^{\mathrm{\infty }}$ topology: for every chart $\phi :U\to V\subseteq {\mathbb{R}}^n$, every compact $K\subset V$, and every integer $m\ge 0$, use the seminorm that is the maximum of the absolute values on $K$ of all coordinate coefficients and their partial derivatives of order at most $m$. On a compact manifold these are the usual smooth-form seminorms and no condition at infinity is involved.

A degree-$k$ de Rham current on $X$ is a continuous real-linear functional $T:{\mathrm{\Omega }}^{n-k}(X)\to \mathbb{R}$, for $0\le k\le n$. Its pairing is written $⟨T,\eta ⟩$. Put current spaces outside degrees $0,\dots ,n$ equal to zero. For $k<n$, define the current differential by $⟨dT,\eta ⟩=(-1{)}^{k+1}⟨T,d\eta ⟩,\eta \in {\mathrm{\Omega }}^{n-k-1}(X),$ where $d$ is the ordinary exterior derivative on smooth forms; set $dT=0$ for $k=n$.

For a smooth $k$-form $\alpha$, its associated smooth-form current is $T_{\alpha }(\eta )={\int }_X\alpha \wedge \eta$. If $i:\mathrm{\Sigma }\hookrightarrow X$ is a compact oriented embedded smooth submanifold of dimension $p$ without boundary, its integration current is the degree-$(n-p)$ functional $⟨{\delta }_{\mathrm{\Sigma }},\eta ⟩:={\int }_{\mathrm{\Sigma }}{i}^{\ast }\eta ,\eta \in {\mathrm{\Omega }}^p(X)\mathrm{.}$ For a point $y\in X$, put $⟨{\delta }_y,f⟩=f(y)$ for $f\in {\mathrm{\Omega }}^0(X)$. For finitely many points $y_a$ and real coefficients $q_a$, the finite point-source current is $\rho ={\sum }_a{q}_a{\delta }_{y_a}$ and its total coefficient is $⟨\rho ,1⟩={\sum }_a{q}_a$.

The symbol $d$ on smooth forms is the exterior derivative specified by the invariant vector-field formula: for an $m$-form $\omega$ and smooth vector fields $Y_0,\dots ,Y_m$, $d\omega (Y_0,\dots ,Y_m)={\sum }_{j=0}^m(-1{)}^j{Y}_j(\omega (Y_0,\dots ,{\widehat{Y}}_j,\dots ,Y_m))+{\sum }_{0\le i<j\le m}(-1{)}^{i+j}\omega ([Y_i,Y_j],Y_0,\dots ,{\widehat{Y}}_i,\dots ,{\widehat{Y}}_j,\dots ,Y_m)\mathrm{.}$

For a smooth map $F:M\to N$, the pullback of $\omega \in {\mathrm{\Omega }}^r(N)$ is defined pointwise by $(F^{\ast }\omega {)}_x(v_1,\dots ,v_r)={\omega }_{F(x)}(d{F}_x{v}_1,\dots ,d{F}_x{v}_r)$; in the integration-current formula $F=i:\mathrm{\Sigma }\hookrightarrow X$ is the inclusion.

Every integral here uses the finite chart-localized oriented integral. For a compactly supported top form $\zeta$ on an oriented smooth manifold $M$, take finitely many compactly supported smooth cutoffs ${\chi }_j$ in oriented chart domains whose sum is $1$ near $\mathrm{supp}\zeta$, and sum the signed Euclidean chart integrals of ${\chi }_j\zeta$. The resulting integral is independent of this finite localization and chart choices; for $\dim M\ge 1$, it satisfies ${\int }_{M}d\lambda =0$ for compactly supported $(\dim M-1)$-forms $\lambda$. Since $X$ and $\mathrm{\Sigma }$ are compact, the top forms integrated here have compact support.

Smooth manifolds and their smooth charts

Oriented smooth manifolds and oriented charts

Embedded smooth submanifolds with boundary

A smooth differential $k$-form

The exterior derivative by the invariant vector-field formula

The pullback of a differential form

Pullback of forms is smooth functorial and preserves wedges

Finite chart localization gives choice-free integration and compact Stokes

The integration-current continuity and closedness, the continuity of the current differential, and the compatibility of $d$ with smooth-form currents are proved by the items in the justified_by field. A closed oriented submanifold is thus represented by a closed current. The term “Poincaré-dual current” is conventional shorthand for this integration current; the definition alone does not identify current cohomology with singular cohomology.

These are mathematical currents with real coefficients. The numbers $q_a$ are only mathematical coefficients; this definition assigns no physical units, brane charge, Wess–Zumino normalization, or source sign.