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Given a space X with non-trivial homology, we can modify X into Y, e.g., by capping some n-boundaries, so that maps of f to itselt induce isomorphisms only on, say, $H_1(X)$, but not on $H_i(X)$ for $i>1$ . What are examples of maps from a space X to itself ( by space , I mean a combination of the points with a topology, to avoid changes like the capping of boundaries) that have the same effect, i.e., maps $f:X\rightarrow X$ that induce isomorphisms on , say, $H_i(X)$, for $i=1,2,\ldots,j$, but do not induce isomorphisms for $j+1,\ldots,n$?

Thanks.

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A bland example: A constant map from the sphere $S^{j-1}$ to itself. Note here that for $k > j$, the homologies are again isomorphic(being trivial). –  George Sep 8 '11 at 13:47
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This question is very open ended. Is there something in partiulcar you'd like to see? –  Jason DeVito Sep 8 '11 at 15:34
    
Maybe some nice examples , where the maps are not constant and where the isomorphisms are not trivial, i.e., from trivial-to-trivial. –  gary Sep 8 '11 at 16:13
    
It would also be nice to know if there are some related theorems/results. –  gary Sep 8 '11 at 16:57

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