# Is there a general theory of fiber theorems?

Here are three vague theorems rolled up in one.

Let $X$ and $Y$ be sufficiently nice topological spaces and $f:X \to Y$ a sufficiently nice surjection. If for each $y \in Y$, the fiber $f^{-1}(y) \subset X$ is (topologically, homotopically, homologously) equivalent to a point, then $f$ induces a (topological, homotopic, homological) equivalence between $X$ and $Y$.

There are many such theorems (Vietoris-Begle for homology, Smale for homotopy), and generalizations abound. For instance, instead of requiring fibers to be homotopy-equivalent to points, you can ask that they be $n$-connected for some $n$; under this weaker hypotheses one can still extract the fact that the homotopy groups of $X$ and $Y$ agree up to dimension $n-1$.

This suggests a fairly general template: take a nice map $f:X \to Y$, and assume that each fiber $f^{-1}(y)$ is equivalent to a point under [insert topological invariant $I$ here]. Then, $X$ and $Y$ are $I$-equivalent.

Question: Is there a general theorem like the one suggested above?

For instance, one might ask: if each fiber has the Euler characteristic of a point, must we obtain $\chi(X) = \chi(Y)$? The reason I thought to ask is because I would like to know if such a theorem holds for simple homotopy equivalence (see here for instance). Even if nothing like the super-general theorem above can hold, I'd still be interested in knowing if one has the following precise, restricted version:

Let $X$ and $Y$ be connected CW complexes and $f:X \to Y$ a cellular surjection. If for each $y \in Y$, the fiber $f^{-1}(y) \subset X$ is simple homotopy equivalent to a point, then $X$ and $Y$ are simple homotopy equivalent.

If this is true, where can one find this theorem in the literature? I scoured Marshall Cohen's book on Simple Homotopy Theory but could not find it.

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It is a theorem of Marshall Cohen that a PL map of finite polyhedra is a simple homotopy equivalence if it has contractible point-inverses. This was in a 1967 Annals paper entitled "Simplicial structures and transverse cellularity". – Allen Hatcher Mar 19 '13 at 3:47
Two related results have to do with cell-like maps. A cell-like map is one in which the fibres are cell-like spaces. These are compact spaces which are contractible in a very strong sense. Result 1: A cell-like map is exactly a map $f:X\to Y$ such that any restriction $f:f^{-1}(U)\to U$ is a proper homotopy equivalence for any $U$ open in $Y$. Result 2: even more interestingly, cell-like maps between several types of manifold-like spaces are approximable by (and homotopic to) homeomorphisms. See mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0111.0128.ocr.pdf for a nice overview. – Ricardo Andrade Mar 19 '13 at 7:36
A nice result along the lines of your first boxed theorem is Dror's Theorem H.1 in "Cellular spaces, null spaces and homotopy localization". It says that, given a homotopy fibration $F\rightarrow E \rightarrow X$, if $L$ is a localization functor and $L(F)=*$ then $L(E)\rightarrow L(X)$ is an equivalence. You recover the homotopic and homological cases in this way. – Fernando Muro Mar 19 '13 at 8:40
@Fernando: I think the question is also referring to situations where the maps are not necessarily fibrations of any kind. – Ricardo Andrade Mar 19 '13 at 9:35
For simplicial complexes, there are the poset fiber theorems, whch somehow fit within your general statement: ams.org/journals/tran/2005-357-05/S0002-9947-04-03496-8 – Camilo Sarmiento Mar 19 '13 at 18:09

Edit: I have added some definitions and details to my answer.

In the most general form I can find, your third question is a consequence of two results regarding cell-like maps and fine homotopy equivalences. It is closely related to and makes use of Chapman's celebrated theorem on topological invariance of Whitehead torsion.

Firstly, a cell-like map is a map which is proper, and whose fibres are all cell-like spaces. A compact metric space is called cell-like if it is "shape contractible" in the sense of having trivial (Borsuk) shape. Secondly, a map $f:X\to Y$ is a fine homotopy equivalence if, for any open cover $\mathcal{U}$ of $Y$, $f$ has a homotopy inverse $g$, and corresponding homotopies to the two identity maps which both follow paths never leaving some open in $\mathcal{U}$. Finally, recall that ANR abbreviates absolute neighbourhood retract. Importantly, and to avoid piling up qualifiers, all ANRs in this answer will be implicitly assumed metric and separable. A good example of a (metric separable) ANR is a locally finite, countable CW-complex.

The main theorem in the article "Mappings between ANRs that are fine homotopy equivalences" by William Haver implies that any cell-like map between locally compact ANRs is a (proper) fine homotopy equivalence. In that article, a cell-like map would be called a proper $UV^\infty$-map instead.

On the other hand, corollary 3.2 in the article "The homeomorphism group of a compact Hilbert cube manifold is an ANR" by Steve Ferry implies that any proper fine homotopy equivalence between locally compact ANRs is a simple homotopy equivalence. The proof of this result by Ferry uses the theory of Hilbert cube manifolds, via the following facts: (1) a proper fine homotopy equivalence between Hilbert cube manifolds is approximable by homeomorphisms (as proved by Ferry in the article cited above), and (2) the product of a locally compact ANR with the Hilbert cube is a Hilbert cube manifold (by a result of Robert Edwards). Finally, it uses Chapman's results on the topological invariance of Whitehead torsion.

In any case, putting together the two results stated above, we conclude the following.

Any cell-like map between locally finite, countable CW-complexes is a simple homotopy equivalence.

To relate this to the actual question, first observe that simple homotopy type is not defined for every space. In the greatest generality I am aware of, it is definable for locally compact ANRs. Then being simple homotopy equivalent to a point is equivalent to being a compact contractible ANR. Since such a space is necessarily cell-like, we obtain the following answer to your last question.

Let $f:X\to Y$ be a proper map between locally finite, countable CW-complexes. Assume each fibre of $f$ is a contractible ANR, i.e. has trivial simple homotopy type (since the fibres are compact). Then $f$ is a cell-like map. By the above result, $f$ is a simple homotopy equivalence.

That is the most general statement I can find. If you only care about the case of finite CW-complexes, then the result admits a formulation with fewer classifiers, and was also proved a few years earlier. For example, it is stated explicitly as a theorem in page 17 of the article by Lacher cited below.

If $f:X\to Y$ is a cell-like map between compact ANRs, then $f$ is a simple homotopy equivalence. In particular, if $f:X\to Y$ is a map between finite CW-complexes whose fibres are all contractible ANRs, then $f$ is a simple homotopy equivalence.

Finally, two nice overviews of the theory of cell-like maps and related topics are:

Moreover, the first part of the article by Lacher contains an interesting exposition of theorems of Vietoris-Begle type, much like the ones mentioned in the question. It might be a good place to start looking for a general theory of such results.

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Thanks! This definitely answers the concrete question! – Vidit Nanda Mar 19 '13 at 22:23
@Vel Nias: My pleasure. I admit this stuff is way beyond my expertise, but the references above should go some way towards explaining what I described in my answer. Finally, I would like to renew my suggestion to look at the first part of Lacher's article. It might give you some leads regarding your more general question. – Ricardo Andrade Mar 19 '13 at 23:08
Ricardo, I'm definitely going to take a look at Lacher's paper. Thank you again. – Vidit Nanda Mar 20 '13 at 1:18