A metric Kan-Thurston theorem
For every simplicial complex , we construct a locally CAT(0) cubical complex , a cellular isometric involution on and a map with the following properties: ; is a homology isomorphism; the induced map from the quotient space to is a homotopy equivalence; the induced map from the fixed point space to is a homology isomorphism. The construction is functorial in .
One corollary is an equivariant Kan-Thurston theorem: every connected proper -CW-complex has the same equivariant homology as the classifying space for proper actions, , of some other group . From this we obtain extensions of a theorem of Quillen on the spectrum of a (Borel) equivariant cohomology ring and a result of Block concerning assembly conjectures. Another corollary of our main result is that there can be no algorithm to decide whether a CAT(0) cubical group is generated by torsion.
In appendices we prove some foundational results concerning cubical complexes, including the infinite dimensional case. We characterize the cubical complexes for which the natural metric is complete; we establish Gromov’s criterion for infinite-dimensional cubical complexes; we show that every CAT(0) cube complex is cubical; we deduce that the second cubical subdivision of any locally CAT(0) cube complex is cubical.
remark[theorem]Remark \newnumberedquestion[theorem]Question \newnumberedexamples[theorem]Examples \newnumbereddefinition[theorem]Definition \classno20F67 (primary), 20J06, 55P99, 57S30 (secondary) \extralinePartially supported by NSF grant DMS-0804226 and by the Heilbronn Institute. Some of this work was done at MSRI, Berkeley, where research is supported by NSF grant DMS-0441170.
In , D. Kan and W. Thurston showed that every simplicial complex has the same homology as some aspherical space , or equivalently every connected simplicial complex has the same homology as the classifying space for some discrete group . More precisely, they give a map which induces a homology isomorphism for any local coefficient system on . Furthermore, the construction of and has good naturality properties. The Kan-Thurston theorem has been extended by a number of authors [3, 13, 22]. In particular, Baumslag Dyer and Heller showed that in the case when is finite, the space may be taken to be a finite simplicial complex . (In contrast, the original was uncountable whenever had dimension at least two.)
The Kan-Thurston theorem has also been generalized in a number of ways. McDuff showed that every has the homotopy type of the classifying space of some discrete monoid . Leary and Nucinkis showed that every has the homotopy type of a space of the form for some discrete group , where denotes the classifying space for proper actions of .
Our purpose is to give new proofs of some of these results using methods that come from metric geometry. We give an extremely short summary of these methods before stating our result. A good general reference for this material is .
A geodesic metric space is a metric space in which any two points are joined by a ‘geodesic’, i.e., a path of length equal to their distance. A geodesic metric space is CAT(0) if any geodesic triangle is at least as thin as a comparison triangle in the Euclidean plane having the same side lengths. Any CAT(0) space is contractible: for any point one may define a contraction of to by moving points at constant speed along the (unique) geodesic path joining them to .
Let be a cubical complex. The link of any vertex in is naturally a simplicial complex, and vertex links in the universal cover are isomorphic to vertex links in . There is a metric on in which distances are defined in terms of the lengths of piecewise-linear paths, using an isomorphism between each cube of and a standard Euclidean cube of side length one to measure the length of paths. Gromov has given a simple criterion for this metric to be CAT(0): this happens if and only if is simply connected and the link of each vertex is a flag complex [8, 11]. In particular, it follows that if all vertex links in are flag complexes, then is aspherical.
The standard references for Gromov’s criterion consider only the case when is finite-dimensional (see [8, theorem II.5.20], [11, section 4.2.C], [7, ch. 2] or the unpublished  for example). In Theorem B.6 in an appendix to this paper we give a proof in the general case.
Let be a locally CAT(0) cube complex. There are many known constraints on the fundamental group of such a . For example, if is finite, then is bi-automatic, which implies that there are good algorithms to solve the word and conjugacy problems in . As another example, when is countable, the group has the Haagerup property, i.e., admits a proper affine isometric action on a Hilbert space. This implies that can contain no infinite subgroup having Kazhdan’s property (T) .
In contrast, we show that there is no restriction whatsoever on the homology of a locally CAT(0) cubical complex. We also show that there is no restriction on the homotopy type of the quotient of a locally CAT(0) cubical complex by an isometric cellular involution, apart from the obvious condition that it should be homotopy equivalent to a CW-complex. The following statement is a version of our main theorem.
[A] Let be a simplicial complex. There is a locally CAT(0) cubical complex and a map with the following properties.
The map induces an isomorphism on homology for any local coefficients on .
There is an isometric cellular involution on so that and the induced map is a homotopy equivalence.
The map induces an isomorphism on homology for any local coefficients on , where denotes the fixed point set in for the action of .
If is finite, then so is . The construction is functorial in the following senses:
Any simplicial map that is injective on each simplex gives rise to a -equivariant cubical map .
In the case when is injective, embeds isometrically as a totally geodesic subcomplex of .
In the case when is locally injective, is a locally isometric map. In particular, in this case is injective on fundamental groups.
If is the union of subcomplexes , then is equal to the union of the . The dimension of is equal to the dimension of , except that when is 2-dimensional, is 3-dimensional. In all cases, the dimension of is equal to the dimension of .
By a locally injective map of simplicial complexes, we mean a map which is injective on each simplex of , and for each vertex induces an injective map from the link of to the link of . Equivalently, a locally injective map of simplicial complexes is a simplicial map which induces a locally injective map of realizations.
The existence of and having property 1 is a strengthening of the Baumslag-Dyer-Heller version of the Kan-Thurston theorem [3, 16], since the only groups used are fundamental groups of locally CAT(0) cubical complexes. Similarly, the existence of and having property 2 is a strengthening of the theorem of Leary and Nucinkis on possible homotopy types of since the only groups used are groups that act with stabilizers of order one and two on CAT(0) cubical complexes . This follows from the fact that whenever a group acts with finite stabilizers on a CAT(0) cubical complex that cubical complex is a model for (see Theorem B.9).
We describe a number of corollaries to Theorem A in Sections 8–11 of this paper. We believe that none of these results follow from earlier versions of the Kan-Thurston theorem, and we hope that they go some way towards motivating our work.
One advantage of Theorem A over other versions of the Kan-Thurston theorem is that it readily yields an equivariant version, which we state as Theorem 8.3. If acts on which we suppose to be connected, then acts also on . Let be the universal covering space of , and let be the group of self-homeomorphisms of that lift the action of on . If we assume that acts on with finite stabilizers, then acts with finite stabilizers on the CAT(0) cubical complex . It follows (see Theorem B.9) that is a model for , the classifying space for proper actions of . By construction, there is a surjective group homomorphism , and an equivariant map which is an equivariant homology isomorphism.
Our construction of and having property 1 is closely modelled on the construction of groups used by Maunder in  and by Baumslag, Dyer and Heller in . The main ingredients in the proof in  can be illustrated by solving a simpler problem: suppose that one wants to construct, for each , a group so that the integral homology of is isomorphic to the homology of the -sphere. To solve this problem, they fix an acyclic group , together with an element of infinite order. Take to be the subgroup of generated by , and define groups (to play the role of the -disc) and for as follows. Let , our fixed acyclic group. Now define by gluing two 2-discs along a circle: . Define , and check that is an acyclic group containing as a subgroup. (The standard copy of and its conjugate by together generate a subgroup of isomorphic to .) The definitions of the higher groups are now clear: and .
This construction carries over without change to the metric world, once one has found an acyclic locally CAT(0) cubical complex to replace the classifying space for . The proof from  when translated into the metric world gives a construction of the space denoted by in the statement of Theorem A. To obtain and the involution , we need a second acyclic locally CAT(0) cubical complex , containing and equipped with an involution such that is equal to the fixed point set and such that the quotient space is contractible. Roughly speaking, is constructed by attaching a copy of to every copy of that appears in the construction of . The acyclic locally CAT(0) 2-complex that we construct is a presentation 2-complex, in which the 2-cells are large, right-angled polygons which in turn are made from smaller squares.
The remainder of this paper is structured as follows. In Section 2, we consider the problem of making a CAT(0) polygon with given side lengths from unit squares. In Section 3 we build acyclic locally CAT(0) 2-complexes that will be used in the construction of and . Section 4 constructs , and as described in the previous paragraph, and Section 5 gives the proof of Theorem A, following .
Sections 6 and 7 consider variants on the construction of , including a cube complex whose cubical subdivision is , a version which is always metrically complete but for which the map is not necessarily proper, and a version defined for arbitrary spaces , and such that the map from to is a weak homotopy equivalence. Section 8 contains Theorem 8.3, an equivariant Kan-Thurston theorem for actions with finite stabilizers. Sections 9 and 10 discuss applications of Theorem 8.3 to Borel equivariant cohomology and to assembly conjectures. In particular, Corollary 9.2 is an extension to Quillen’s theorem on the spectrum of a Borel equivariant cohomology ring , and Theorem 10.1 generalizes an observation of Block [5, Introduction]. Block’s observation concerned the Baum-Connes conjecture for torsion-free groups, while our Theorem 10.1 applies to groups with torsion and to many assembly conjectures. In Section 11 we use Theorem A to deduce that there can be no algorithm to decide whether a CAT(0) cubical group is generated by torsion. Section 12 discusses some open questions related to our work.
Sections A, B and C contain results about cubical complexes with an emphasis on the infinite-dimensional case, and should be viewed as appendices to the paper. In some cases the results for finite-dimensional cubical complexes are well-known, and our proofs proceed by reducing the arbitrary case to the finite-dimensional case. In Theorem A.5 we characterize the cubical (and simplicial) complexes for which the natural path metric is complete. In Theorem B.6 we establish Gromov’s criterion: a simply-connected cubical complex is CAT(0) if and only if all links are flag complexes. In Theorem B.9 we show that whenever a finite group acts on a CAT(0) cubical complex (not assumed to be metrically complete), the fixed point set is non-empty. Section C concerns the comparison between cubical complexes and the more general complexes often called cube complexes. The result is that in the CAT(0) case there is almost no advantage to using cube complexes rather than cubical complexes. In Theorem C.2, we show that every CAT(0) cube complex is in fact cubical. In Corollary C.7 we deduce that the second cubical subdivision of every locally CAT(0) cube complex is cubical.
The results in Sections A, B and C, will not be surprising to experts, but we know of no published proof the main results that covers the infinite-dimensional case. In a private communication in 2007, Michah Sageev told the author that Yael Algom-Kfir had obtained a proof of Gromov’s criterion for infinite-dimensional cube complexes (proved here as Theorem B.6) while working on her MSc with him. Also in 2007, Yael Algom-Kfir told the author that she did not intend to publish her proof. Our proof of Theorem B.6 involves a reduction to the case of a finite-dimensional cubical complex, and we rely heavily on results of Sageev  in making this reduction.
Acknowledgements.This work was started in 2001, when the author visited Wolfgang Lück in Münster and Jean-Claude Hausmann in Geneva. Some progress was made at MSRI in 2007, and the author thanks these people and institutes for their hospitality. The author also thanks many other people for helpful conversations concerning this work, especially Graham Niblo, Boris Okun and Michah Sageev. Finally, the author thanks Raeyong Kim and Chris Hruska for their helpful comments on earlier version of this paper.
2 Tesselating CAT(0) polygons
We seek CAT(0) tesselations by squares of polygons with given side lengths. A tesselated -gon is defined to be a 2-dimensional cubical complex which is homeomorphic to a disc, together with an ordered sequence of distinct vertices on the boundary circle of . It will be convenient to view the index set for these vertices as rather than , so that . Call the vertices the corners of the polygon, and call the complementary pieces of the (open) sides of the polygon. The th side is the piece whose closure contains and . The th side length is defined to be the number of edges in the th side.
Suppose that is a tesselated polygon. For a vertex of , let denote the degree of , i.e., the number of neighbouring vertices. Define the curvature of at to be
We say that is a CAT(0) tesselated polygon if for every vertex , . By Gromov’s criterion, it follows that the natural path-metric on is a CAT(0) metric in which each of the sides of is a geodesic.
An easily proved combinatorial version of the Gauss-Bonnet theorem shows that for any -gon ,
It follows easily that the only tesselated CAT(0) quadrilaterals are rectangles with side lengths , since a CAT(0) quadrilateral must have for all . Similarly, a tesselated CAT(0) pentagon can have for just one vertex .
Clearly, some restrictions are required on the side lengths for the construction of a CAT(0) -gon. For example, the perimeter , defined by must be even since every edge not in belongs to exactly two squares. Note also it cannot be the case that any is greater than or equal to , since each side of the polygon is a geodesic. We do not characterize the sequences for which exists, but just give some constructions that suffice for our purposes.
First we give three ways to obtain a new CAT(0) polygon, starting from a CAT(0) -gon with side lengths .
One can introduce a new corner in the middle of the th side of the -gon, to produce an -gon: if satisfies then an -gon with side lengths is obtained.
One can add a rectangle to the th side of the -gon, producing a new -gon with side lengths . By applying this process once for each , one obtains an -gon with side lengths consisting of the original -gon with a ‘collar’ attached.
One can take the cubical subdivision of an -gon, producing an -gon with side lengths double those of the original -gon.
Next we discuss two ways to build a CAT(0) polygon, starting just from the required side lengths.
Suppose that is a Euclidean rectangle tesselated by squares, with perimeter and corners . Any choice of vertices in which gives the required side lengths gives a way to view as a tesselated -gon. However, this -gon will fail to be CAT(0) if . Define subrectangles of as follows: if there exists such that , then let be the single point ; if lies in the interior of the th side of , let be the subrectangle of with , and as three of its vertices. Provided that the four rectangles are disjoint, we define to be the closure in of . This is a CAT(0) -gon with the required side lengths. Note that the curvature of is concentrated along , since each interior vertex of has valency 4.
For example, consider the problem of realizing a pentagon with side lengths . If we start with a square of side 2, there are two distinct ways to cut up the boundary. One of these does not work, since two of the subrectangles and corresponding to adjacent vertices meet. The other way gives a valid in the form of an ‘L’-shape consisting of three squares.
As a second example, consider the problem of realizing a hexagon with sides . Again, there are two ways to break up the boundary of a square of side 2, one of which gives , with four of the hexagon’s vertices coincident with vertices of and two on the midpoints of opposite edges. The other way does not work, since for some the subrectangles and meet, and so the construction gives a bow-tie consisting of two squares joined at a vertex. The defect with this ‘hexagon’ is that the midpoints of the two long sides coincide. There is however another way to realize the hexagon with sides , as a suitably labelled rectangle.
As a third example, Figure 1 illustrates a CAT(0) octagon with seven sides of length 2 and one side of length 4 that was constructed in this way. Note that the curvature in this octagon is concentrated at two of its corners and at the midpoints of two of its sides.
One corollary of this construction is that every sequence with divisible by 4 and each strictly less than can be realized: the condition on the lengths guarantees that if we choose to be a square, then the four rectangles are disjoint.
Next consider the problem of constructing a CAT(0) -gon in which all the curvature is concentrated at a single vertex. In the case when the vertex with non-zero curvature is in the interior of , this is equivalent to identifying the sides of rectangles where has side lengths and . (As usual the index should be assumed to lie in the integers modulo .) This gives rise to a CAT(0) -gon with side lengths given by the equations . The cases when the vertex with non-zero curvature is in arise as degenerate cases. If but for , then the vertex with non-zero curvature lies in the interior of the th side of the -gon. If but for , then the curvature is concentrated at the corner of . Other degenerate cases do not give rise to CAT(0) -gons.
The equations can be solved uniquely for if and only if is not divisible by 4. If , there is a solution if and only if
If we define by in the case when is odd, and when is congruent to 2 modulo 4, then the unique solution in these cases is given by
At least in the case when 4 does not divide , this solves completely the question of whether there exists a CAT(0) -gon with curvature at just one vertex and side lengths . The above equation computes , and then there is such an -gon if and only if each is a positive integer, and either at most one is equal to zero or there exists such that if and only if or . Since every CAT(0) pentagon has curvature at just one vertex, this completely solves the question of the possible side lengths of CAT(0) pentagons.
Figure 2 shows three CAT(0) hexagons that can be built in this way by identifying the sides of six Euclidean rectangles.
3 Acyclic 2-complexes
We give some constructions for acyclic locally CAT(0) 2-complexes. In each case we start with a balanced finite group presentation. We take a rose in which each petal has the same length, and attach right-angled CAT(0) 2-cells as constructed in Section 2. Since we wish to build our right-angled polygons using unit squares, the lengths of the petals of the roses that we use will be larger; petals of length 4 will suffice.
Fix integers and with , and construct a 2-complex by attaching polygons with sides to a rose with petals. The petals of the rose will be labelled and for . The sides of the polygons will be labelled not by a single letter, but instead by a longer word. To describe the process it is convenient to let stand for a reduced word in the letters and their inverses which begins and ends with . For example, is a valid choice for , whereas is not. Similarly, let denote a reduced word in the letters and their inverses which begins and ends with . Define a meeting point to be a pair consisting of a word and its formal inverse , where and , so that there are distinct meeting points. The sides of the polygons will be labelled by words and of the types described above, or their inverses, in such a way that at the meeting points appearing at the corners of the polygons are all distinct. It is clear that there are many ways to do this. Two such labellings will suffice for our purposes, the words
for or the words
for . In each case, the different appearances of each and should be read as representing possibly different words of the allowed type, rather than lots of copies of the same word.
The subwords and occurring in the words that represent the boundaries of the polygons should be chosen in such a way that there is a tesselated CAT(0) polygon whose side lengths are four times the word lengths and . For example, if and all of the words and are of approximately equal lengths, then such tesselated CAT(0) polygons can always be made by cutting the corners off a Euclidean rectangle as described in the previous section.
The 2-complex is built by starting with a rose with petals, each built from four 1-cells. Each petal is labelled by one of the letters or . Now the tesselated CAT(0) polygons are attached using the words described above. To check that this 2-complex is locally CAT(0), we check the link of each vertex. To simplify the proof, we consider only the case when the curvature in each of the polygons is concentrated at interior vertices. The links of vertices in the interiors of the polygons are circles of circumference at least , and so cause no problems. Similarly, the link of any vertex in , apart from the central vertex of the rose, consists of two points joined by a number of arcs each of length . It remains to consider the central vertex of the rose. The link of this vertex is a graph with vertices, the inward and outward ends of the petals of the rose, which we shall denote by , , and for . There are edges in this graph of length , coming from the corners of the -gons, and a large number of edges of length , coming from the interiors of the sides of the -gons. For each and each , there is at most one short edge between and . Thus the short edges form a subgraph of the complete bipartite graph . Each long edge either has both of its vertices in or both of its vertices in . The fact that the words along the sides of the -gons are reduced implies that no single long edge is a loop. Hence the shortest loops in the link graph consist of either two long edges, one long and two short edges, or four short edges. It follows that the 2-complex is locally CAT(0).
The 2-complex will be acyclic if and only if the abelianization of its fundamental group is trivial. This of course places extra conditions on the words along the edges of the -gons. We state some results that can be proved using this process.
Note that the subcomplex of consisting of the 0-cell and the 1-cells labelled ‘’ is a totally geodesic subcomplex, since the points in the vertex link in are separated by at least . It follows that the subgroup of generated by is a free group on these generators. Similarly, the subgroup of generated by is a free group freely generated by these elements.
The group presented on generators subject to the following 6-relators is non-trivial, torsion-free and acyclic. The corresponding presentation 2-complex may be realized as a non-positively curved square complex (or as a negatively curved polygonal complex).
Here , and we have written in place of and similarly in place of . Each word and consists of a power of a single letter. Using techniques from Section 2, one shows easily that CAT(0) hexagons with side lengths , and can be built from unit squares, and these suffice to make this 2-complex: in fact, the cubical subdivisions of the three hexagons shown in Figure 2 can be used for this purpose. The claim in parentheses follows from the fact that obtuse hexagons with side lengths in these ratios can be realized in the hyperbolic plane.
There is a non-contractible finite locally CAT(0) acyclic 2-dimensional cubical complex which admits a cubical involution such that
The fixed point set consists of a single point;
The quotient space is contractible;
contains an isometrically embedded -invariant 2-petalled rose with acting by swapping the two petals.
Let , and for define
These words and will be fixed throughout this proof. Now consider the words
The techniques of Section 2 enable one to construct a CAT(0) octagon with each side length 24 except for one side of length 28. For example, one could take the CAT(0) tesselated octagon depicted in Figure 1, and collar it five times to produce a CAT(0) octagon with seven sides of length 12 and one of length 14. The cubical subdivision of this octagon is the octagon that we require. Attach 8 copies of this octagon to an 8-petalled rose (with petals labelled by the 8 generators) to make the 2-complex .
There is an action of on the eight generators, with acting by sending to and to . This extends to a free action of on the eight octagons of . We define the involution to be the action of the element , so that and . The fixed point set for the action of on is just the central vertex of the rose, and so is a single point as claimed. Let be the quotient map. There is a natural cell structure on with one 0-cell and four 1- and 2-cells. Since and , the words describing the attaching maps for the four 2-cells are no longer reduced. After reduction they are equal to , , and . Thus the 2-complex is homotopy equivalent to a wedge of four 2-discs, and so is contractible.
The subcomplex of the rose consisting of the base vertex and the petals labelled and is a 2-petalled -invariant rose which is isometrically embedded in .
Here are two other families of group presentations that can be shown to arise as fundamental groups of finite locally CAT(0) square complexes using similar techniques:
Generators , and relators
Fix , and take generators and relators , for .
4 Contractible and acyclic complexes
For our main result we require a pair of locally CAT(0) cubical complexes and a cubical involution of the pair with the following properties:
and are acyclic
is not contractible
is a totally geodesic subcomplex of
is contained in the fixed point set for acting on
The following proposition gives a fairly simple construction that has almost all of the properties that we require.
Let be an acyclic CW-complex, and let denote the involution of defined by . The quotient space is contractible.
It suffices to show that the fundamental group is trivial and that is acyclic. Since fixes points on the diagonal of , is isomorphic to the quotient of the wreath product by the normal subgroup generated by . Thus we need to show that this wreath product is generated by conjugates of . Recall that elements of the wreath product may be written in the form , where and or . The product of two such elements is given by
The inverse of is , and so for any ,
is a conjugate of , and is a product of conjugates of . In particular, for any and ,
lies in the normal subgroup generated by . But since is perfect, an arbitary element of may be expressed as a product of such commutators. Hence for any , and lie in the normal subgroup generated by , which is therefore the whole of .
Now let denote the diagonal copy of within . Note also that is equal to the fixed point set for the action of on . Since is acyclic, the inclusion of in induces a homology isomorphism. Hence the relative homology groups are all trivial. It follows that the relative cellular chain complex is an exact sequence of free -modules, and hence is split. Thus this sequence remains exact upon tensoring over with (with acting trivially). Now there is a natural isomorphism
and hence the relative homology groups are trivial. From this it follows that the inclusion of in is a homology isomorphism, and hence that is acyclic, as required.
For any prime , a similar argument may be used to show that is contractible, where acts on by freely permuting the factors.
The only problem with the above construction is that if is an acyclic locally CAT(0) cubical complex, the diagonal copy of inside is not a subcomplex. Nevertheless, the barycentric subdivision of the diagonal copy of is a subcomplex of the barycentric subdivision of .
Proposition 4.1 can be used to prove a weaker version of Theorem A, in which is not a locally CAT(0) cubical complex, but instead a locally CAT(0) polyhedral complex. In the non-metric context, Proposition 4.1 is also useful. It can be used to give an alternative proof of the main theorem of , starting from a proof of the Kan-Thurston theorem along the lines of that of Baumslag-Dyer-Heller .
There is a pair of locally CAT(0) cubical complexes equipped with an involution having the properties listed at the start of this section. Furthermore is the fixed point set for the action of . The complex is 2-dimensional and is 3-dimensional.
Let be the 2-complex constructed in Proposition 3.2, and let denote the involution of which is denoted ‘’ in Proposition 3.2. Let be the direct product of with an interval of length 8, and define an involution on by . By definition, can be viewed as being built from unit cubes. Define to be the mapping torus of , i.e., the quotient of obtained by identifying and for all . Since is an involution, it follows readily that defines an involution on . Let denote the unique 0-cell in , and define by identifying the images of the points and in . As before, passes to an involution on . Each of , and is a locally CAT(0) cubical complex, as may be seen by checking that all vertex links are flag. (Every vertex link in or is isomorphic either the cone on or the suspension of a vertex link in , and the ‘new’ link in is isomorphic to the disjoint union of two links from .)
Let be image in of , so that is a 2-petalled rose in which each petal has length 4. The subspace is totally geodesic and invariant under the action of (which swaps the two petals). Let be a disjoint copy of , and let be the subspace of consisting of the 0-cell and the two 1-cells labelled and . Then is also a 2-petalled rose in which each petal has length 4, is totally geodesic in and is invariant under the action of the involution , which swaps the two petals. Let be an isometry which is equivariant for the two given involutions, and define by taking the disjoint union of and and identifying with for all . Since was constructed from two locally CAT(0) cubical complexes by identifying isometric totally geodesic subcomplexes, it follows that is itself a locally CAT(0) cubical complex. (Alternatively, one may check the structure of vertex links in .) Since has the property that for all , one may define an involution on by for and if . Let be the fixed point set for . It is readily seen that is equal to the image of inside , which is isometric to and so is acyclic. The spaces , , and are depicted in Figure 3. The fixed point set for the involution on each is shaded in grey.
It remains to check that is acyclic and is contractible. Since is acyclic, it follows easily that has the same homology as a circle. A Mayer-Vietoris argument now shows that the inclusion map is a homology isomorphism. Since is acyclic, another Mayer-Vietoris argument shows that is acyclic.
Note that is homeomorphic to the mapping cylinder of the quotient map . Since is contractible, it follows that is contractible. From this it follows that is homotopy equivalent to a circle, and that the inclusion map is a homotopy equivalence. Since is contractible, it follows from van Kampen’s theorem that has trivial fundamental group and it follows from the Mayer-Vietoris theorem that is acyclic. Hence is contractible as claimed.
5 The main result
Our proof is close to Maunder’s proof of the Kan-Thurston theorem . However, we use a slightly wider class of spaces than Maunder’s class of ‘ordered simplicial complexes’. We start with the category of -complexes, or semi-simplicial sets [12, 30]. Our construction will apply to the full subcategory of -complexes such that the edges of each -simplex are distinct.
Note that the barycentric subdivision of any -complex has the stronger property that the vertices of each -simplex are distinct, and so the barycentric subdivision of any -complex is in . Similarly, the barycentric subdivision of any simplicial complex is naturally an object of , and any map of simplicial complexes that is injective on each simplex induces a map of -complexes from to .
Let and denote -complexes, and let be a vertex of . The link of , denoted , is another -complex. Any map of -complexes induces a map . (This is one advantage of -complexes over simplicial complexes.) The map is said to be locally injective if for each vertex , the map is injective. It can be shown that a map of -complexes is locally injective if and only if the induced map of topological realizations is locally injective.
For each in , we will construct a locally CAT(0) cubical complex , a map and an involution on having the properties stated in Theorem A. The construction will be natural for any map in ; however the induced map will be locally isometric only in the case when is locally injective. The following statement is a summary of what we will prove.
Let denote the category whose objects are locally CAT(0) cubical complexes equipped with a cellular isometric involution and whose morphisms are -equivariant cubical maps. There is a functor from to which has all of the properties listed in Theorem A.
Theorem A follows from Theorem 5.1 by composing with the barycentric subdivision functor from the category of simplicial complexes and simplicial maps that are injective on each simplex to the category .
Before beginning the proof of Theorem 5.1, we need two more pieces of notation. If is a locally isometric cellular map of cubical complexes, and is a positive integer, let denote the mapping cylinder of of length , i.e., the cubical complex obtained from by identifying, for each , the points and .
Finally, let be the pair of spaces constructed in Theorem 4.2, and let denote a fixed locally isometric closed loop . For example, could be the map that goes at constant speed around the generator for the fundamental group of . (Recall that is isometric to the cubical complex of Proposition 3.2.)
For each finite -complex in , we inductively define , the map , another locally CAT(0) cubical complex , a locally isometric cubical map and cubical involutions on , , so that all of the following hold.
The map induces integral homology isomorphisms and and for any vertex of , an isomorphism of fundamental groups .
is acyclic and is contractible.
The map is -equivariant.
and are finite.
, except that if then .
except that if , and if is 0-dimensional.
If is a subcomplex of , then is a totally geodesic subcomplex of and is a totally geodesic subcomplex of , and these inclusions are -equivariant.
In the case when is 0-dimensional, define to be , define to be a single point, take to be the unique map, and let be the trivial involution.
In the case when is 1-dimensional, take to be the second barycentric subdivision of , so that for each edge of , is either a loop or a path consisting of 4 edges of length 1. Let be the pair of spaces constructed in Theorem 4.2, and for take the 1-point union of copies of the space indexed by the 1-cells of , with the involution as defined in Theorem 4.2. The map is defined by taking a copy of the map for each 1-cell.
Now suppose that , , and have been defined and have the above listed properties whenever is finite and consists of at most simplices of various dimensions. Suppose now that is obtained from by adding a single -simplex , for some . Let denote the subcomplex of consisting of all the simplices contained in the boundary of . To simplify notation slightly, let denote , let denote the -fixed point set , and let denote the restriction to of the map . Now define
More precisely, let be the initial vertex of , and let be the corresponding vertex of . Note that is fixed by the involution on . Also let be the fixed point for the involution of described in Proposition 3.2. Now the space is obtained from the disjoint union by identifying with and identifying with .
The involution on each of and is defined using the given involutions on , , , and . The map is defined to equal on . On the map is equal to , and on the rest of the mapping cylinder it is induced by the map from to . The map is defined by on , by taking to the barycentre of , and on the rest of the mapping cylinder by linear interpolation between the map and the constant map that sends to the barycentre of .
It follows easily that the maps and are isometric embeddings of totally geodesic subcomplexes. The quickest way to see this is by induction, using a gluing lemma, but one could also show directly that each vertex link in (resp. ) is a full subcomplex of the corresponding vertex link in (resp. ). Similarly, an induction shows that the map is a -equivariant locally isometric cubical map.
It is easily checked that the map induces isomorphisms from each of , and to . By the 5-lemma and induction it follows that induces isomorphisms from , and to .
By induction, the fundamental group of is trivial, and induces an isomorphism , for any vertex . By Van Kampen’s theorem, it follows that induces an isomorphism from to as claimed.
The Mayer-Vietoris theorem shows that , and so by induction is acyclic. Similarly, Van Kampen’s theorem and an induction shows that is simply-connected. It follows that is contractible as required. This completes the proof that the construction applies to any finite and has the properties listed at the start of the proof in this case.
For the general case, define , and as direct limits over the finite subcomplexes of . (We shall not need or in the case when is infinite, but these could also be defined in this way for arbitrary .) It is immediate from the definition that in the general case, induces isomorphisms from , and to and an isomorphism, for any , from to .
To establish the metric properties of in the general case, it is easiest to use Gromov’s criterion (Theorem B.6). If is a vertex of , then the link of , , is the limit of the , where ranges over the finite subcomplexes of such that contains . Since each such