Let $\alpha$ be a differential form on the torus $T^n$ whose support $\mathrm{supp}(\alpha)$ is contained in a small neighborhood of the subtorus $T^k\equiv T^k\times \{0\}^{n-k}$.



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> **Question:**
> Suppose $\alpha$ is closed or even harmonic with respect to some metric. I was wondering if the de Rham cohomology class $[\alpha]\in H^\*\_{dR}(T^n)$ has to live in the image of the pullback $H^\*\_{dR}(T^k)\to H^\*\_{dR}(T^n)$ induced by the projection $T^n\to T^k$.
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Actually I firstly thought about the following question: if $S$ is a singular chain/cycle whose image is contained in a small neighborhood of $T^k$, then do we have $[S]\in H\_\*(T^n)$ must lie in the image of $H\_\*(T^k)\to H\_\*(T^n)$? The answer for this should be positive as we can continuously retract $S$ into $T^k$. But in cohomology theory as above, I get confused.


For simplicity one may assume $k=1$ and $n=2$. For more generality, we may consider a pair of (compact) smooth manifolds $N\subset M$ rather than the torus $T^k\subset T^n$.



A differential form whose support is in a tubular neighborhood of $T^k\times \{0\}^{n-k}\subset T^n$
By compactness, $\operatorname{supp}(\alpha)\subset T^k\times B^{n-k}$,
where $B^{n-k}\subset T^{n-k}$ is a small open ball.
So $[\alpha]$ is in the image of $H^\*\_{dR}(T^n,T^n\setminus T^k\times B^{n-k})\to H^\*\_{dR}(T^n)$.
By the Künneth formula and excision,
$$ H^\*\_{dR}(T^n,T^n\setminus T^k\times B^{n-k})
\cong H^\*\_{dR}(T^k)\otimes H^\*\_{dR}(\overline{B^{n-k}},\partial B^{n-k})\;.$$
The second factor only has cohomology in degree $n-k$, generated by, say $[\omega]$. The image of $[\omega]$ in $H^{n-k}(T^{n-k})$ is a generator as well. So there exists a unique $\beta\in H^\*\_{dR}(T^k)$ such that
$$[\alpha]=[\beta]\otimes[\omega]\;.$$


More generally, let $N\subset M$ both be compact and let $N$ have orientable normal bundle $\nu$. If $U\subset M$ is a tubular neighbourhood of $N$ with $\operatorname{supp}(\alpha)\subset U$, then $U$ is diffeomorphic to $\nu$ and $[\alpha]$ is in the image of
$$H^\*\_{dR}(N)\stackrel\Theta\longrightarrow H^\*\_{dR}(M,M\setminus U)\longrightarrow H^\*\_{dR}(M)\;,$$
where $\Theta$ is the Thom isomorphism for the normal bundle (followed by excision).
This composition is sometimes denoted $\iota\_!$ ($\iota\colon N\to M$ is the inclusion). It raises degree by the rank of the normal bundle.


If both $N$ and $M$ are oriented, then so is $\nu$, and one can describe $\iota\_!$ by conjugating the push forward $\iota\_\*$ in homology with Poincaré duality on $N$ and $M$.


