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Let $X$ be a path-connected topological space, let $x, x' \in H_{k}(X)$ for $k>0$ be represented by two connected manifold i.e. there exist two compact oriented connected manifolds $M$, $N$ and two continuous maps $f \colon M \rightarrow X$, $g \colon N \rightarrow X$ such that $f_{\ast}[M]=x$ and $g_{\ast}[N]=x'$ (here $[M]$ and $[N]$ denote the fundamental classes of the respective manifolds). I have to prove that even the sum $x+x'$ in $H_k(X)$ can be represented by a connected manifold.

Thanks in advance for any help.

N.B.
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  • This fails for $k=0$ and $X$ nonconnected. – archipelago May 28 '15 at 20:01
  • Thanks: editing the question to ignore this cases. – N.B. May 28 '15 at 20:03
  • I think $X$ should be also path-connected. Then you could try to work with the connected sum of the manifolds. – Daniel Valenzuela May 28 '15 at 23:49
  • The connected sum of the manifolds was my first idea: using the long exact sequence in homology for the pair $(M \sharp N, S^{k-1})$ I have the injective map $H_{k}(M \sharp N) \rightarrow H_k(M \sharp N, S^{k-1})\cong H_k(M \vee N) \cong H_k(M) \oplus H_k(N)$. I suppose this map sends $[M \sharp N]$ to $[M]+ [N]$ but then I must find a function $M \vee N \rightarrow X$ taking $[N] + [M]$ into $x+x'$. If the maps $f,g$ were pointed I think I can use the induced map $f \vee g \colon M \vee N \rightarrow X$ but I don't have this assumption. – N.B. May 29 '15 at 08:05
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    This might interest you. – Najib Idrissi May 29 '15 at 10:08

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