Cylindrical algebraic decomposition (abbreviated c.a.d.; see Definition 3.1) is a method that decomposes a semi-algebraic subset into simpler pieces (cells) in a systematic way. It first arose  in the context of quantifier elimination, but has since become a useful technique for effective computation of topological invariants, such as homology groups, of semi-algebraic sets. For example, the piano movers’ problem (see, inter alia, ) asks whether the configuration space of allowable positions of an object in a subset of is connected. Questions of this nature can have both theoretical and practical importance. 111Some of the results of this paper formed part of the Bath Ph.D. thesis  of the second author, which was funded by the University of Bath. We acknowledge discussions with Matthew England and David Wilson, and EPSRC grant EP/J003247/1 which funded them. GKS thanks Andrew Ranicki and Kenichi Ohshika for education about cobordism.
Cell decompositions can be quite pathological, however, and for purposes of computation (again, both theoretical and practical) some further conditions are needed. One would hope, at least, to obtain a representation of as a CW-complex: better still, a regular cell complex.
Let be a closed and bounded semi-algebraic set.
Can we find a c.a.d. of into regular cells?
Given a c.a.d. of , can we tell easily whether it is a regular cell decomposition?
A partial answer to Question 1.1(i) was given in , where it is shown that the bounded cells of a semi-monotone c.a.d. (see ) are regular, and an algorithm is given to construct such a c.a.d. if or . However, being semi-monotone is a strong condition and it is not at present clear whether semi-monotone c.a.d.s exist at all in general. Even if they do, they are likely to be laborious to construct and to have many cells, making them unsuitable for computation.
We can always find a c.a.d. of with regular cells if we allow a change of coordinates, but this is usually undesirable. From a computational point of view, implementations of c.a.d. algorithms often improve run time by exploiting sparseness, which is destroyed by change of coordinates. Quantifier elimination, the original motivating example for c.a.d. in , does not allow arbitrary changes of coordinates, and indeed requires some ordering on the coordinates: we assume, as is usual in c.a.d. theory, a total order . Thus it is important to understand which c.a.d.s have good properties such as giving regular cell decompositions.
For these reasons the earlier study of the topological properties of c.a.d.s in  remains very relevant to Question 1.1. Lazard describes some much weaker conditions and conjectures (Conjecture 3.13, below) that a c.a.d. satisfying them will have regular cells, and also shows how to construct these c.a.d.s for .
Much earlier, Schwartz and Sharir  had proved that a c.a.d. produced by Collins’ algorithm , the only method known at that time, gives a regular cell complex provided it is well-based (see Definition 3.8).
In Section 3.1 of this paper we prove that any well-based strong c.a.d gives a regular cell complex: see Theorem 3.11 for the precise statement and Definition 3.6 for the meaning of “strong”. A well-based c.a.d. produced by Collins’ algorithm is always strong, so this is a generalisation of the result of Schwartz and Sharir, but it is entirely independent of the method used to construct the c.a.d. and is thus more widely applicable.
These two results are superficially similar but quite different in detail. In Section 3.1 we consider a c.a.d. that is -invariant (see Definition 3.3) for a large set of polynomials , including as a minimum all the polynomials that are used to define . (Indeed, the term well-based itself already presumes that is -invariant.) In one respect, this restriction is not onerous: algorithms commonly do produce -invariant c.a.d.s by construction. On the one hand, -invariance is a very strong global condition, which may force to have many cells even far from and cannot be checked locally near each cell.
By contrast, in Section 3.2 we are concerned with the topology of c.a.d.s in general, subject only to local conditions. Apart from its theoretical interest, this is potentially important in the context of Brown’s NuCAD algorithm , which constructs cells that are capable of being cells in c.a.d.s, rather than complete c.a.d.s, and is thus inherently local: global conditions such as -invariance do not arise.
This difference is also reflected in the methods of proof of Theorem 3.11 and Theorem 3.27. For Theorem 3.11, we use largely elementary methods of real algebraic geometry, exploiting the rigidity imposed by the -invariance. The tools used to prove Theorem 3.27 are topological and are anything but elementary as they include the -cobordism theorem (in effect, the Poincaré Conjecture).
Some of the statements make sense over an arbitrary real closed field, but as our methods are in part topological we work over throughout. See  for an approach to -cobordism in the context of real closed fields, which could possibly allow one to remove this restriction.
A subsidiary aim of this paper is to give some consistent terminology for ideas that have appeared in different parts of the literature under various, sometimes incompatible, names. We try to do this in the course of Section 2, which explains the background to the problems. The main results are found in Section 3. Finally, in Section 4, we make some brief observations about another question (Question 4.2) raised by Lazard in the same paper .
2 Cells and cell decompositions
Throughout the rest of the paper, we use and to denote the open ball and the sphere, respectively, in with centre and radius (the dimension will always be clear): we use and for the standard unit ball and sphere in . If then denotes the closure of in the Euclidean topology.
We begin with a well-known example, which motivates Question 1.1.
Put and consider the semi-algebraic set
a subset of the Whitney umbrella .
We can partition into nine disjoint cells: the corners , , and ; the edges , and (for ) together with (for ); and .
2.1 Definitions and terminology
Next, we collect some definitions. This is already not quite trivial, because the same or very similar conditions have been introduced by several authors at different times under very different, and sometimes incompatible, names. Before doing any mathematics at all, we propose some terminology which we believe is consistent, flexible and memorable.
A subset of is a -cell, for , if there exists a homeomorphism , for some called the dimension of . The boundary of (sometimes, for emphasis, the cell boundary) is .
The cell boundary of a cell does not, in general, coincide with the topological boundary of , which is . Also, may have a structure of manifold with boundary in which the manifold boundary (see Definition 3.15) might coincide with neither nor the topological boundary.
Let be a subset of . A cell decomposition of is a partition of into disjoint cells.
Even in a cell may have bad boundary: for instance, if we take then has infinitely many connected components. We define below some desirable conditions on a cell and its boundary. Some of these conditions are intrinsic to ; others are related to a cell decomposition.
Recall that if and are inclusions of topological spaces, a homeomorphism is a homeomorphism such that is a homeomorphism .
We say that two subsets and are equiregular if there exists a homeomorphism . A -cell is said to be a regular cell if is equiregular with .
The hypercube is a regular cell. On the other hand, the cell is not regular, even though its closure is . Moreover, even if is regular, a particular homeomorphism need not extend to even as a continuous map.
We establish a convention for naming cell decompositions where all cells have a certain property.
If is a property of cells we shall say that is a cell decomposition if is a cell decomposition and all cells of satisfy .
Thus a regular cell decomposition is a decomposition into regular cells. Not every property of can be checked on the cells, however: for example, a finite cell decomposition is simply a decomposition into finitely many cells. There is no ambiguity, because finiteness is not a property of cells.
We say that two cells and in are adjacent if either one intersects the closure of the other. We say that is subadjacent to , written , if .
If and , then by [3, Theorem 5.42], .
Now we define some extrinsic properties of a cell, in relation to a cell decomposition.
Let be a cell of a cell decomposition . We say that is closure finite in if (or, equivalently, ) is the union of finitely many cells of .
This condition is found in the literature under different names. It is called boundary coherent in [15, Definition 2.7]: elsewhere, sometimes without the finiteness requirement, it is called the frontier condition. We use the term closure finite as it is more descriptive than either of the terms above, and is the usual term in the topology literature. Indeed, it is the meaning of the “C” of “CW complex”: see [13, p. 520].
2.2 Examples and basic properties
We first illustrate some relations among the properties introduced in Section 2.1.
Consider the following cell decomposition of , in which the end points and are not cells, and the edges and () are not sub-divided by these points.
The cube is closure finite; the -cell , subadjacent to the cube, is not closure finite.
Example 2.8 also shows that even if is closure finite in its boundary may contain a cell of that is not closure finite.
The cells and in are not closure finite in , and they are subadjacent to each other.
Let be a cell of a cell decomposition . Then is closure finite if and only if implies that . In particular, if two cells and of are closure finite and subadjacent to each other, then .
Suppose that is closure finite in ; that is, . If then it must intersect some , and thus . In particular . Conversely, if contains all cells subadjacent to , then . ∎
Let be a cell of a cell decomposition . We say that is well-bordered in if there is a finite collection of cells of such that and .
For instance a -sphere minus a point is a cell that is not well-bordered. See [15, Example 2.9] for more examples of how a cell can fail to be well-bordered, and for a closure finite decomposition that is not well-bordered.
Like closure finiteness, the well-bordered property does not permeate to subadjacent cells.
Consider the cell decomposition that consists of the open cube , all six of its faces, eleven of its edges (not the -axis) and seven of its corners (not the origin), together with the -cell and the -cell . We observe the following:
The cell is well-bordered but not closure finite.
the faces that are adjacent to the cell are neither well-bordered nor closure finite.
These two conditions are nevertheless related.
Let be a cell of a cell decomposition . If and all cells subadjacent to are well-bordered, then is closure finite.
In view of Lemma 2.10 it suffices to show that if and . We proceed by induction on : for there is nothing to prove.
As is well-bordered, for some finite collection of cells with . If , for some , then ; otherwise, for some . Then by induction and the result follows. ∎
Any well-bordered cell decomposition is closure finite.
As we have seen, [15, Example 2.9] shows that the converse is not true. However, for regular cell decompositions the two coincide.
A regular cell decomposition of a compact set is closure finite if and only if it is well-bordered.
If is a closure finite regular -cell then is homeomorphic to , and decomposes into finitely many cells. Then is the closure of the union of the -cells in that decomposition.
The other direction is just Corollary 2.14. ∎
For any topological property , we say that a set is locally boundary if every has a base of neighbourhoods in such that holds for each .
In many cases one may take the neighbourhoods to be the intersections for : we shall do this without further comment when it is convenient, but one should check that it is permissible to do so. An example of a property for which this would not be permissible is disconnectedness: is locally boundary disconnected (as well as being locally boundary connected!) because we may take for the sets , but the intervals are all connected. We shall not in fact consider any properties for which the balls are not suitable neighbourhoods.
Lazard [15, Definition 2.7] defines “boundary smooth”, which according to Definition 2.16 is the same as “locally boundary connected”. We prefer this terminology because it extends to other properties (we shall need “locally boundary simply connected” later, for instance) and because the term “smooth” is already overloaded. In particular, “boundary smooth” has nothing to do with either being or the absence of singularities.
2.3 Semi-algebraic cell decompositions
Now we limit ourselves to semi-algebraic cells. We shall make constant use of the conic structure of semi-algebraic sets [5, Theorem 9.3.6]. We also need a slightly stronger relative version (take to recover the usual version).
Suppose that are semialgebraic subsets of and . Then for there exists a semi-algebraic homeomorphism which is the identity on , such that for all and is the cone on with vertex .
This is a semi-algebraic set (it is the mapping cylinder of the inclusion ) so we may consider its conic structure near a point where . Then it is sufficient to take to be the restriction to of the map in guaranteed by [5, Theorem 9.3.6]. ∎
The following consequence of the local conic structure is also useful.
Let be a semi-algebraic set and . Then for the intersection has as a deformation retract.
Applying [5, Theorem 9.3.6] to yields a homeomorphism between and the cone on . Away from , this restricts to a homeomorphism between and , and the latter retracts onto . ∎
If is a homotopy property for which Definition 2.16 can be checked on balls, and is a semi-algebraic cell, then is locally boundary if and only if, for all , there exists such that has property for all .
Clearly the same is also true with instead of .
With the definitions we have made, being locally boundary is automatically an equiregularity invariant property. In particular, as was pointed out in , a regular cell, even if not semi-algebraic, is always locally boundary connected.
3 Cylindrical algebraic decomposition
We think of a cylindrical algebraic decomposition as a finite partition of into semi-algebraic cells, built inductively, and whose projections onto the first variables, for , are either disjoint or the same. These cells are not just arbitrary semi-algebraic sets homeomorphic to , for some , but cells that arise from graphs of some semi-algebraic functions , as below.
A cylindrical algebraic decomposition or c.a.d. of is a finite semialgebraic cell decomposition of defined inductively by the following conditions.
If then is any finite cell decomposition of .
The projection on the last coordinates is cylindrical: that is, if then either or .
is a c.a.d. of .
For each there are finitely many continuous semi-algebraic functions , satisfying for all and all , such that for each and : furthermore both and also belong to .
In Definition 3.1, the graphs are called sections and the cells are called sectors of .
A c.a.d. is usually chosen to respect some data, such as some functions on or subsets of .
Let be a finite set of nonzero polynomials. A c.a.d. is said to be -invariant if, for every , the sign of is constant on each .
This is sometimes called sign-invariance: one could instead require other properties of , such as its order of vanishing, to be constant on each , but we shall not need any other kind of invariance here.
Let be a semi-algebraic set. A c.a.d. of is adapted to if is a union of cells of .
It is sometimes useful to give some more information about .
A sampled c.a.d. is a c.a.d. together with a choice of base point for each cell .
In general, a c.a.d. is not a CW complex: for instance, a c.a.d. can fail to be closure finite.
We say that a c.a.d. of is a strong c.a.d. if is well-bordered and locally boundary connected.
3.1 -invariant c.a.d.s
The aim of this section is to investigate when an -invariant c.a.d. adapted to a closed bounded semi-algebraic set exhibits as a regular cell complex.
If is an -invariant c.a.d. and is a section, we put . We say that is reduced if , for every section .
We do not require that should actually be cut out by , but we will usually impose the next condition, which may be seen as a weaker version.
If is an -invariant c.a.d., we say that a section , with , is a bad cell for if there is an such that . If there are no bad cells, we say that is well-based.
Note that the definitions of bad cell and well-based depend on as well as . Note also that we apply the term “bad cell” to , not . This makes no difference, since if is a bad cell then so is any with (consider the same ), but it does reflect our point of view of starting with a given c.a.d. rather than constructing one inductively.
We aim to show that reduced well-based strong c.a.d.s give regular cell complexes (see Theorem 3.11 for the precise statement). This was shown in [20, Theorem 2] for a c.a.d. constructed via Collins’ algorithm : by [15, Theorem 4.4], such a is strong if it is well-based. Other algorithms are now in use, though, such as c.a.d. via regular chains  or via comprehensive Gröbner systems , so we want to be able to dispense with the condition on the construction.
We need two lemmas: the first is a variant of [5, Lemma 2.5.6].
Suppose that is a finite set of non-zero polynomials and denote by
its closure under the operator . Let be a bounded section in an -invariant c.a.d. of , for a semi-algebraic continuous bounded function . If and is well-based, then can be extended continuously to .
The proof is similar to step (ii) in the proof of [5, Lemma 2.5.6]. It is enough to show that extends continuously to , for an arbitrary .
By the Curve Selection Lemma [5, Theorem 2.5.5], we choose a continuous semi-algebraic path , such that and for . Then we define , for : since is bounded by hypothesis, is a bounded continuous semialgebraic function and hence extends continuously to by [5, Proposition 2.5.3].
Now we extend to by putting , and on . The claim is that is continuous at . If not, then
and hence if we define then . Again applying the Curve Selection Lemma we obtain a path with and for , so exactly as before we put and this extends continuously to . By continuity, we have , and also and .
Now suppose that , so and . Consider the polynomial , and observe that . If is not the zero polynomial, we may consider the set of all derivatives of . By -invariance, for any given the sign of is the same near (say at for small ) as near . Hence cannot have opposite signs at and , although one might be zero and the other not.
But this contradicts Thom’s Lemma [5, Proposition 2.5.4]: at two distinct zeros of a real polynomial in one variable, some derivative must have opposite signs.
Hence, if is discontinuous at , we must have : that is, is identically zero above . But then, by cylindricity and -invariance, must be identically zero above the cell in containing , contrary to the assumptions. ∎
Next we need a lemma that allows us to pass extensions up from subdivisions.
Let be a bounded, local boundary connected section in an -invariant c.a.d. of , for a semi-algebraic continuous function . Suppose that is a well-based strong -invariant c.a.d. refining (i.e. each cell in is a subset of a cell of ), in which is partitioned into sections for semi-algebraic continuous functions . If all the extend continuously to , then extends continuously to .
It is enough to show that if for some , then . Then is consistently defined by if .
We first show this with the assumption that . Then, since and therefore are strong and in particular closure finite, we have by Lemma 2.10. Therefore agrees with on (they both agree with ) and hence agrees with also on , which is contained in .
For general and , we construct a subadjacency chain from to , by considering a semi-algebraic path between and ; this is possible as is semi-algebraic and connected, and thus semi-algebraically path-connected by [5, Prop. 2.5.13]. The path gives us a way of selecting the correct consecutive cells.
Let be a semi-algebraic path with and . As semi-algebraic, has finitely many connected components, for any ; thus is a finite collection of intervals contained in .
Considering the preimage of every cell in , we get a finite partition where . Denote by the unique cell that contains . Then , so and are adjacent: moreover belongs to either or , so one is subadjacent to the other.
Now we have a finite chain of not necessarily distinct cells
where each stands for either or . Hence , so is well-defined. ∎
Suppose is an -invariant, reduced, well-based strong c.a.d. of adapted to a closed and bounded subset . Then the corresponding decomposition of is a regular cell complex.
A strong c.a.d. of is closure finite, so we just need to show that is a regular cell. Moreover, if we can show that the sections of are regular, then by [20, Lemma 5], so are the sectors.
We will show that a section is a regular cell by proving that is homeomorphic to , where is the cell below ; then the result follows by induction. It suffices to show that extends continuously to , since then and are mutually inverse homeomorphisms.
Let be the closure of under ; that is, the smallest set that contains and is closed under partial differentiation with respect to . We may choose a -invariant c.a.d. refining . Then is partitioned into sections and each for the continuous semi-algebraic function (with, as usual, ).
Now we apply Lemma 3.9, remembering that since there is an that vanishes on . We conclude that each can be extended continuously to .
Finally, we use Lemma 3.10 to extend continuously to . ∎
The above, in combination with Lemma 2.15, prompts us to raise the following question.
Suppose that is a c.a.d. of : is it true that is closure finite if and only if it is well-bordered?
Some of the results in this section can be strengthened slightly, by weakening global conditions so that they only apply where they are needed. For example, in Theorem 3.11 it would be enough for to give an -invariant c.a.d. of some open cylinder containing rather than of the whole of . Similarly, in Lemma 3.9 it is enough for to be well-based (even just near ), and the condition of -invariance can also be relaxed analogously.
3.2 Topology of strong c.a.d.s
The following conjecture is made in [15, p. 94].
Suppose is a strong c.a.d. of adapted to a closed and bounded semi-algebraic set . Then the corresponding decomposition of is a regular cell complex.
We prove some cases of this using techniques from topology. By the nature of the proofs, they are valid only for , not for arbitrary real closed fields. The idea is that a regular cell is automatically a manifold with boundary: conversely, it is sometimes possible to give conditions on a manifold with boundary that are sufficient to ensure that it is a regular cell, and in low dimension we are able to verify that these conditions always hold.
Let be a compact -manifold. We say that is a homology -sphere if has the same homology groups as . We say that is a homotopy -sphere if has the same homotopy type as .
It follows from the Hurewicz theorem [13, Thm 4.32] and Whitehead’s theorem [13, Thm 4.5] that any simply connected homology sphere is a homotopy sphere. One must heed the warning given in  after the proof of Whitehead’s theorem: we need a weak homotopy equivalence, that is, a map that induces isomorphisms between the homotopy groups. However, if is an open ball then a map that identifies with the complement of a point and sends all of to is a weak homotopy equivalence between and .
A topological space is a -dimensional manifold with boundary if, for each , there exists a neighbourhood of that is homeomorphic to an open set in either or . The manifold boundary of , denoted , is the set of points of with no neighbourhood homeomorphic to an open set in .
If for some we say that is compatibly a manifold with boundary if .
For more on manifolds with boundary see [13, p 252]. Our main use of them is based on the following easy result.
Let be a -dimensional compact, contractible manifold with boundary. Then the boundary of is a homology -sphere.
The homology sequence of is
and Lefschetz duality [13, Theorem 3.43] gives . Since is contractible we have and for , and hence and for . ∎
Suppose that is a proper surjective continuous map between connected, locally compact separable metric spaces, and is locally contractible. If all the fibres of are contractible and locally contractible, then is a weak homotopy equivalence.
We use Theorem 3.17 to deduce the following, which may be of independent interest.
Suppose is a c.a.d. of and the induced c.a.d. of is strong. Then the closure of any bounded cell of is contractible.
As often in Section 3.1, let . We shall show that and have the same homotopy type: then is contractible by induction on . The result is true for , so we assume .
As and are bounded, and are compact semi-algebraic sets and thus, by [5, Thm 9.4.1], and each admits a CW-complex structure. By Whitehead’s theorem, it suffices to show that and are weakly homotopy equivalent, and for that it is enough to verify that satisfies the conditions of Theorem 3.17.
The spaces and are connected as they are the closures of connected spaces. Moreover, as they are also Hausdorff compact metric spaces, they are locally compact and separable. It follows immediately from the local conic structure theorem [5, Thm 9.3.6] that any semi-algebraic set, in particular and every fibre of , is locally contractible. The map is a continuous map between a compact space and a Hausdorff space, so it is closed and proper.
Finally, because is a strong c.a.d., by [15, Proposition 5.2] the fibres of are closed segments and thus contractible. ∎
What conditions on a c.a.d. are necessary to ensure that the closures of its bounded cells are contractible? Could it be true for an arbitrary ?
In fact a compact contractible manifold with boundary is a -cell () if and only if is simply connected. For this one uses the topological -cobordism theorem, which is a consequence of the (generalised) Poincaré conjecture: see  for some comments on this, and also [14, Conjecture 3.5].
More precisely: is contractible so is a homology -sphere by Lemma 3.16, so is a homotopy -sphere because it is simply-connected. Then is cobordant with the sphere bounding a small ball . Because this cobordism is an -cobordism it is homeomorphic to , and we get that is homeomorphic to which is .
Consequently, to show that a c.a.d. of a compact -dimensional semi-algebraic set is a regular cell complex it is enough to show that every cell closure is compatibly a compact contractible manifold with boundary and, for , that is simply connected.
In general, the closure of a c.a.d. cell is not compatibly a manifold with boundary. The cell in Example 2.1 is an example of this: but is strictly contained in . Another example is the non-regular cell mentioned after Definition 2.4.
In very low dimension the position is simple.
If is a locally boundary connected - or -cell of , then is compatibly a manifold with boundary.
This is immediate from the definition of manifold with boundary. Using a more involved argument, we can show that, under certain conditions, closures of -cells are compatibly manifolds with boundary.
A locally compact space is a homology -manifold if, for all
A homology manifold is not a manifold in general, but homology -manifolds are -manifolds for . In some ways, they behave better than manifolds, as the following fact (stated in , and easily checked by the Künneth formula) suggests.
Suppose and are topological spaces. If is a topological manifold, then and are homology manifolds.
If is a semi-algebraic cell and then is a open subset of a manifold and thus a manifold. As in the proof of Proposition 2.18, the local conic structure gives a homeomorphism
so is a homology manifold for .
We can determine when locally boundary connected -cells of a cell decomposition are manifolds with boundary.
Let be a locally boundary connected semi-algebraic -cell in . If, for all and , the set is homeomorphic to , then is compatibly a manifold with boundary.
Therefore , giving a homeomorphism .
Hence by the local conic structure of semi-algebraic sets, is homeomorphic to the cone on , by a map sending to the vertex. This is a manifold with boundary, and , so is a manifold with boundary and . But because is a manifold, so is compatibly a manifold with boundary. ∎
We can apply this to strong c.a.d.s.
Let be a -cell of a strong c.a.d. of . Then is compatibly a manifold with boundary.
By the previous discussion, is a connected -manifold so it is homeomorphic to either or . As a strong c.a.d. is well-bordered, there exists a -cell , such that . If is homeomorphic to , then is the cone with vertex on , but that has an isolated boundary point at . ∎
We have proved the following result.
Let be a -dimensional compact semi-algebraic set. If is a strong c.a.d. adapted to , then represents as a regular cell complex.
To prove Conjecture 3.13 for by this method, we would need to show that -cells of a strong c.a.d. of are manifolds with boundary. We have not been able to do this but we can do so under the additional assumption that the cells of the c.a.d. are locally boundary simply connected: see Definition 2.16.
Let be a strong c.a.d. of . If is a locally boundary simply connected -cell of , then is compatibly a manifold with boundary.
For and , we know that is a homology -manifold and therefore a -manifold.
We claim that