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Suppose I have $X=X_1\cup X_2\cup…\cup X_n$ and $f:X \to Y$ where $Y$ has a similar decomposition.

Suppose I know that $f | X_{i_1}\cap…\cap X_{i_r} \to Y_{i_1}\cap…\cap Y_{i_r}$ is a homotopy equivalence for each nonempty intersection. Suppose that the spaces are nice enough that we have the homotopy extension property wherever we want it.

Then $f$ has a homotopy inverse preserving all of this structure and the homotopies themselves preserve the structure.

The isn't hard to prove by piecing together strong deformation retractions in mapping cylinders, but I'd rather just point to a reference.

For anyone who hasn't seen this stuff before, the base case is that if $f:(X,A) \to (Y,B)$ is a map such that $f:X \to Y $ and $f|:A \to B$ are homotopy equivalences, then $f$ is a homotopy equivalence of pairs. The proof is to form the mapping cylinder, squash the mapping cylinder of f| to it's top, keeping the whole top fixed, and then squash the rest. Of course, all of the spaces should be nice enough for the homotopy extension theorem to be available.

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Can we express $X$ as a homotopy colimit of the various intersections and the map $f$ as a pointwise homotopy equivalence of diagrams? – Jeff Strom May 8 '13 at 21:41
I do not know of any references for this particular fact. However, as Jeff Strom indicates, if you have "sufficient" homotopy extension properties, then the diagram formed by the intersections of the $X_i$'s (respectively, the $Y_i$'s) should be cofibrant, and thus its colimit is equivalent to its homotopy colimit. Then the homotopy invariance of homotopy colimits in the Hurewicz/Strøm model structure on topological spaces shows that the map induced on the colimits (which are hopefully $X$ and $Y$) is a homotopy equivalence. The book by Hirschhorn contains these model categorical facts. – Ricardo Andrade May 8 '13 at 22:45
@Jeff, Ricardo: I don't think that is the question. I think the data being asked for is a map $g:Y \to X$ sending $Y_i$ into $X_i$ and homotopies $fg \to id_Y$ sending $Y_i$ into $Y_i$ for all time and $gf \to id_X$ sending $X_i$ into $X_i$ for all time. – Oscar Randal-Williams May 9 '13 at 8:09
@Oscar: Thank you for the clarification. I completely misunderstood the question. – Ricardo Andrade May 9 '13 at 9:30
When you say "for each nonempty intersection", it's not immediately clear whether by "nonempty" you mean that the intersection of the $X_{i_j}$ is nonempty or that $r>0$. But clearly (looking at your base case) you cannot mean the latter. And if the former then you seem to be forgetting that an empty intersection of $X_i$s might map to a nonempty intersection of the corresponding $Y_i$s. Why say "nonempty" at all? – Tom Goodwillie May 9 '13 at 11:46

I think an answer is in

tom Dieck, Tammo Partitions of unity in homotopy theory. Composito Math. 23 (1971), 159–167.

With regard to the result on pairs given by Steve, it could be useful to note that the book Topology and Groupoids gives a result in 7.4.2(Addendum) which gives control over the homotopies involved. The utility of this is that it gives a key to one proof of a gluing theorem for homotopy equivalences, which was first given in the 1968 edition of this book and is applied by tom Dieck in his paper. The Addendum is as follows:

We are dealing with the situation $f:(X,X^0) \to (Y,Y^0)$ where each pair has the HEP.

Let $g^{0} :Y_{0} \to X_{0}$ be any homotopy inverse of $f^{0}$ and let $ H^0: f^0g^0 \simeq 1, K^0: g^0f^0 \simeq 1$ be homotopies. Then $g^0$ extends to a homotopy inverse $g$ of $f$ such that the homotopy $fg \simeq 1 $ extends $H^0$ while the homotopy $gf \simeq 1$ extends the sum $$ K^{0} + g^{0}H^{0}f^{0} - g^{0}f^{0}K^{0} $$ of the homotopies $$ g^{0}f^{0} = g^{0}f^{0}1_{X_{0}} \simeq g^{0}f^{0}g^{0}f^{0} \simeq g^{0}1_{Y_{0}}f^{0} \simeq 1_{X_{0}} $$ determined by $H^0,K^0$.

I do not know of a counterexample to the idea of avoiding the kind of "conjugation" given above (though it has been given in the dual situation).

Note that this Addendum easily gives a gluing theorem for $n$ subspaces with a common intersection.

I'll add that the idea for this result came from generalising the proof that a homotopy equivalence of spaces (not necessarily base point preserving) induces an isomorphism of homotopy groups.

My memory is that another paper relevant to the question, but to which I do not have easy access, is

Spanier, E. H.; Whitehead, J. H. C. The theory of carriers and S-theory. Algebraic geometry and topology. A symposium in honor of S. Lefschetz, pp. 330–360. Princeton University Press, Princeton, N.J., 1957.

particularly the work on carriers.

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Proposition 2.7 of my 1990 Mathematische Zeitschrift paper with Michael Weiss Chain complexes and assembly gives the corresponding result for projective chain complexes. The reference in the proof to "Proposition 2.5 of [17]", my 1985 Math. Scand. paper The algebraic theory of the finiteness obstruction, should have been to "Proposition 1.1 of [17]". This is just the well-known result that a chain map is a chain equivalence if and only if the algebraic mapping cone is chain contractible, the analogue of the well-known result that a map of CW complexes $f:X\to Y$ is a homotopy equivalence if and only if $X$ is a deformation retract of the mapping cylinder.

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