In Adams, J.F. *Infinite Loop Spaces* Princ. Univ. Press. page 9 he states Alexander duality theorem 


**Theorem:[Alexander Duality]** $$ H^r(X,G)=H\_{n-r+1}(S^n-X,G)$$


for finite CW-complexes with a "nice embedding". That is to say, $S^n-X$ has a CW-complex $Y$ as deformation retract and $X$ is a deformation retract of $S^n-Y$.


My first question is the following: **Do you know if every CW-complex admits a nice embedding into some $S^n$?** if so, **could you give a reference?**


Thanks in advance.



Alexander duality theorem for CW-complexes and stable homotopy theory
1) The answer is yes, at least up to homotopy. This can be found in Wall's paper:


Wall, C. T. C., Classification problems in differential topology---IV. Thickenings. Topology, 1966, 5, 73–94.


Wall argues this in a cell-by-cell way, making use of transversality.


2) Alternatively, if X is a finite simplicial complex, then one can find an actual embedding by general position arguments. See Rourke and Sanderson's book for this. Note it is classical that every finite CW complex is homotopy equivalent to a finite simplicial complex.


