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I wonder if the following statement is true

Infinite dimensional CW complexes with nontrivial reduced homology are not homotopy equivalent to finite dimensional CW-complexes.

If this is true, is it possible to show it without using higher homotopy group theory? For example, showing that there are nontrivial (co)homology in large enough dimensions?

By "with nontrivial reduced homology", I ruled out the cases like $S^\infty$, which is contractible.

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No, this is not true. For instance, you could take something like $S^1\vee S^\infty$ which is homotopy equivalent to $S^1$.

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