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Consider two CW complexes A and B. Let f be a continuous map from A to B. Take a cone of f and denote it by Cone(f). If homology complex of Cone(f) is acyclic, one can identify homologies of A and B. My question is: is there some similar statement if one works with simple homotopy type? More precisely, if homology complex of Cone(f) is simple homotopy equivalent to the trivial complex, can one say that homology complexes of A and B are simple homotopy equivalent, or something similar ?

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    $\begingroup$ What is the homology complex of a CW complex? What is simple homotopy equivalence for (presumably chain) complexes? $\endgroup$ – Fernando Muro Mar 29 '16 at 22:07
  • $\begingroup$ The only reasonable interpretation of this question that I can see is "if the mapping cone of $f:A \to B$ is simple homotopy equivalent to a point, is it true that $A$ is simple homotopy equivalent to $B$?". $\endgroup$ – Vidit Nanda Mar 30 '16 at 19:45

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