Euler characteristics of collapsing Alexandrov spaces

In this paper we consider the following problem. We fix an upper bound \(n\) for dimension and a lower bound \(\kappa \) for sectional curvature.

An Alexandrov space is a metric space with a lower sectional curvature bound, introduced by Burago-Gromov-Perelman [6] . The motivation for the above problem stems from the well-known fact that the family of \(n\)-dimensional Riemannian manifolds with sectional curvature \(\ge \kappa \) and diameter \(\le D\) is precompact in the Gromov-Hausdorff topology. The limit objects are Alexandrov spaces with curvature \(\ge \kappa \) and dimension \(\le n\). More generally, the family of Alexandrov spaces with curvature \(\ge \kappa \), diameter \(\le D\), and dimension \(\le n\) is compact in the Gromov-Hausdorff topology. Therefore, in principle, the solution to the above problem allows us to understand the topology of spaces in these (pre)compact families by covering them by a finite number of neighborhoods of limit spaces.

In the case \(k=n\), Perelman's stability theorem [20] (cf. [15] ) solves the above problem completely, that is, \(M\) is homeomorphic to \(X\). Hence we will consider the case \(k<n\), called collapse. Although collapsing in low dimensions has been well studied ( [25] , [27] , [18] , [3] ), so far there is no theory in general dimensions. The few exceptions are special cases when \(X\) satisfies some regularity conditions ( [26] , [22] ), where some fibration structures of \(M\) over \(X\) are obtained. In general, it is expected that \(M\) admits some singular fibration structure over \(X\), where the singular fibers arise over the singular strata of \(X\). See also [7] and the references therein for collapsing under two-sided bounds on sectional curvature.

Recently Alesker [1] proposed some conjectures on collapsing Riemannian manifolds and Alexandrov spaces in terms of their intrinsic volumes (also known as Lipschitz-Killing curvatures). For a closed \(n\)-dimensional Riemannian manifold \(M\), the \(i\)-th intrinsic volume, denoted by \(V_i(M)\) (\(0\le i\le n\)), is a geometric quantity defined as follows (see [1] for the precise definition and references). We first embed \(M\) isometrically into Euclidean space (by the Nash embedding, for example) and consider the volume of its \(\varepsilon \)-neighborhood. Then it is a polynomial in \(\varepsilon \ll 1\), and its coefficients, after appropriate normalization, turn out to be independent of the embedding, which are called the intrinsic volumes of \(M\). In fact they can be defined directly in terms of integrals of the Riemann curvature tensor of \(M\). For example, \(V_n(M)\) is the volume of \(M\), \(V_0(M)\) is the Euler characteristic of \(M\), and \(V_{n-2}(M)\) is proportional to the integral of the scalar curvature of \(M\) (and \(V_i(M)\) vanishes if \(n-i\) is odd).

The following is a brief summary of part of Alesker's conjecture.

Here the strata of \(X\) mean (the main parts of) primitive extremal subsets in the sense of Perelman-Petrunin [23] . This stratification reflects both the geometric and topological structures of Alexandrov spaces, and is very closely related to the collapsing phenomena. Note that the number of such strata is finite and that the above quantity \(V_i(E)\) has not yet been defined. It also should be mentioned that the existence of the expected function \(F\) was stated in [1], Theorem 4.6 as an unpublished result of Petrunin. See [1] for more details and further conjectures.

At present only the following results are known for Conjecture 2 .

• For \(V_n\), the volume: the claim follows from the volume convergence theorem of Burago-Gromov-Perelman [6], Theorem 10.8 .
• For \(V_0\), the Euler characteristic: (1) follows from Perelman's stability theorem [20] (cf. [15] ) and (2a) follows from Gromov's Betti number theorem [14] (cf. [17] , [28] ).
• For \(V_{n-2}\), the total scalar curvature: (1) was recently proved by Lebedeva-Petrunin [16], Subcorollary 1.3 and (2a) was proved by Petrunin [24] .
• The special case of a Riemannian submersion was verified by Alesker [2] .
• The \(2\)-dimensional case was verified by Alesker-Katz-Prosanov [3] .

In this paper we prove (2b) of Conjecture 2 for the \(0\)-th intrinsic volume, i.e., the Euler characteristic, not only for Riemannian manifolds but also for Alexandrov spaces. We denote by \(\chi \) the Euler characteristic and by \(\chi _c\) the Euler characteristic with compact support, where the coefficient field of singular cohomology is fixed and omitted.

For a regular (= nonsingular) fibration, the Euler characteristic of the total space splits into the product of those of the fiber and the base space. Therefore, the above formula can be interpreted as capturing the expected singular fibration structure of \(M\) over \(X\), at least at the level of the Euler characteristic.

This paper is actually a continuation (or addendum) of [11] , where the author proved Theorem 3 in the case when \(X\) has no proper extremal subsets. Indeed the geometric ingredients needed for the proof have already been obtained in the previous paper, and what we do here is a purely topological argument. In [11] the author combined Perelman's Serre fibration theorem [22] with the good covering method developed by Mitsuishi-Yamaguchi [19] . This makes it possible to compute the cohomology of \(M\) via a spectral sequence associated with a presheaf on a good covering of \(X\). This presheaf is in a sense constant on the strata of \(X\), and hence the Euler characteristic splits on each stratum as in the case of a regular fibration. This reduces the computation of the Euler characteristic of \(M\) to that of \(X\) in terms of its strata. The latter is done with the help of another result of the author [12] , which showed that each stratum has a deformation retract neighborhood.

Organization. In Section 2 we fix notation and recall some results from [11] and [12] . In Section 3 we prove Theorem 3 . In Appendix A , to complete the proof of Alesker's conjecture, we construct the integer valued function \(F\) on the strata of the limit space asserted in Conjecture 2 (2b). As already mentioned, this is an unpublished result of Anton Petrunin.

Acknowledgment. I would like to thank Professor Semyon Alesker for his interest in my work.

The reader is assumed to be familiar with the theory of Alexandrov spaces and extremal subsets (see [11], Section 3 for a brief summary). Here we fix notation and recall some results from [11] and [12] .

Let \(X\) and \(M\) be Alexandrov spaces as in Problem 1 that are \(\mu \)-close in the Gromov-Hausdorff distance. Note that \(X\) is a fixed space whereas \(M\) is a variable space depending on the choice of \(\mu \). We fix a \(\mu \)-approximation between \(X\) and \(M\) and use the hat symbol \(\hat \ \) to indicate lifts from \(X\) to \(M\). For example, for \(p\in X\), we denote by \(\hat p\in M\) a point that is \(\mu \)-close to \(p\) under this approximation.

Recall that the distance function from \(p\in X\) is called regular at \(x\in X\setminus \{p\}\) if \(\tilde \angle pxy>\pi /2\) for some \(y\in X\), where \(\tilde \angle \) denotes the comparison angle. Note that there exists a neighborhood of \(p\) on which the distance function from \(p\) is regular except at \(p\). We denote by \(\bar B(p,r)\) the closed \(r\)-ball around \(p\).

Let \(E\) be an extremal subset of \(X\) (see [11], Section 3C for extremal subsets). By definition, \(E\) is closed under the gradient flow of any semiconcave function. The family of extremal subsets is closed under taking union, intersection, and closure of difference. The number of extremal subsets in \(X\) is finite. In fact, it is uniformly bounded in terms of dimension, a lower curvature bound, and an upper diameter bound ( [1], Theorem 4.5 , [10], Theorem 1.1(1) ).

Recall that \(E\) is called primitive if it cannot be represented as a union of two proper extremal subsets. For a primitive extremal subset \(E\), its main part \(\mathring E\) is defined as the relative complement of all proper extremal subsets in \(E\). Note that \(p\in \mathring E\) if and only if \(E\) is the minimal extremal subset containing \(p\), that is, the intersection of all extremal subsets containing \(p\), which is primitive. It is known that \(\mathring E\) is a topological manifold. Therefore the main parts of all primitive extremal subsets define a stratification of \(X\) ( [23], Section 3.8 ).

The next lemma shows that regular fibers are constant on each stratum of \(X\).

Note that the choice of \(\mu \) depends on \(r\) and \(s\), and hence on \(p\) and \(q\). This lemma allows us to define the notion of a regular fiber over \(E\) up to homotopy equivalence, which appeared in Theorem 3 . Strictly speaking, we choose some fiber data \((p,r)\) such that \(p\in \mathring E\) for each primitive extremal subset \(E\) of \(X\), and then take \(\mu \) to be sufficiently small so that the conclusion of the theorem holds.

Let \(\{U_\alpha \}_{\alpha =1}^N\) be a good covering of \(X\) (see [11], Section 3D for good coverings). Recall that each \(U_\alpha \) is a superlevel set of a strictly concave function constructed from distance functions. For any nonempty subset \(A\subset \{1,\dots ,N\}\), we denote by \(U_A\) the intersection of all \(U_\alpha \) such that \(\alpha \in A\). We also denote by \(I\) the set of \(A\) such that \(U_A\) is nonempty. For any \(A\in I\), \(U_A\) is contractible by gradient flows of semiconcave functions. In particular it is a good cover in the topological sense. Since \(U_A\) is defined by distance functions, it can be lifted to \(M\), denoted by \(\hat U_A\).

Let \(E\) be an extremal subset of \(X\) (not necessarily primitive). We denote by \(I_E\) the set of \(A\in I\) such that \(U_A\) intersects \(E\). Moreover, if \(E\) is primitive, we denote by \(\mathring I_E\) the set of \(A\in I_E\) such that \(U_A\) does not intersect \(E\setminus \mathring E\). Since \(F=E\setminus \mathring E\) is also extremal, we see that \(A\in \mathring I_E\) if and only if \(E\) is the minimal extremal subset intersecting \(U_A\). Clearly \(I_E\) is the disjoint union of \(\mathring I_E\) and \(I_F\).

For any extremal subset \(E\) of \(X\), the restricted cover \(\{U_\alpha \cap E\}_{\alpha \in I_E}\) is also good in the topological sense. This follows from the fact that any gradient flow preserves extremal subsets. Note that \(A=\{\alpha _1,\dots ,\alpha _k\}\in I_E\) if and only if \(A\in I\) and \(\{\alpha _i\}\in I_E\) for any \(1\le i\le k\) (see the paragraph after the proof of [11], Lemma 5.14 ). In other words, \(I_E\) defines a full subcomplex of \(I\) in the associated nerve, but we will not use this fact (cf. Proposition 13 ).

The following proposition is the key ingredient in the proof of Theorem 3 .

We also use the following fact from [12] .

The case \(E=X\) was proved in [12], Theorem 1.5 . Since the deformation retraction of this theorem was given by a gradient flow which preserves extremal subsets, the general case follows by restricting it to \(E\).

In this section we prove Theorem 3 . We refer to [4], Chapter III for the basic theory of spectral sequences.

Proof of Theorem 3 .

Let \(X\) and \(M\) be as in Problem 1 . Let \(\{U_\alpha \}_{\alpha =1}^N\) be a good covering of \(X\). Let \(I^p\) be the set of \(A\in I\) with cardinality \(p+1\) (see Section 2 for the definition of \(I\)). We denote by \(S^q(U)\) the module of singular \(q\)-cochains on \(U\), where the coefficient field is fixed and omitted.

We consider the Mayer-Vietoris double complex \(\{\prod _{A\in I^p}S^q(\hat U_A)\}_{p,q\ge 0}\), that is, the rows have the Čech coboundary operator (alternating sum of restrictions) and the columns have the singular coboundary operator. The spectral sequence associated with this double complex converges to the cohomology of \(M\).

Let \(\mathcal E\) denote the set of primitive extremal subsets in \(X\). For each \(E\in \mathcal E\), we take fiber data \((p_E,r_E)\) such that \(p_E\in \mathring E\) and denote its regular fiber by \(F_E\). Note that \(I\) is the disjoint union of \(\mathring I_E\) for all \(E\in \mathcal E\) (see Section 2 for the definition of \(\mathring I_E\)). Let \(\mathring I^p_E\) be the intersection of \(I^p\) and \(\mathring I_E\). Since taking homology does not change the Euler characteristic, by looking at the \(E_1\) term of the spectral sequence, we get

\[\begin {aligned} \chi (M)&=\sum _{p,q\ge 0}\sum _{A\in I^p}(-1)^{p+q}\dim H^q(\hat U_A)\\ &=\sum _{p,q\ge 0}\sum _{E\in \mathcal E}\sum _{A\in \mathring I_E^p}(-1)^{p+q}\dim H^q(F_E){\text { \ due to Proposition \ref {prop:good}}}\\ &=\sum _{E\in \mathcal E}\left (\sum _{q\ge 0}(-1)^q\dim H^q(F_E)\cdot \sum _{p\ge 0}\sum _{A\in \mathring I_E^p}(-1)^p\right )\\ &=\sum _{E\in \mathcal E}\left (\chi (F_E)\cdot \sum _{p\ge 0}\sum _{A\in \mathring I_E^p}(-1)^p\right ). \end {aligned}\]

Hence it remains to show that

\[\sum _{p\ge 0}\sum _{A\in \mathring I_E^p}(-1)^p=\chi _c(\mathring E).\]

The left hand side \(\sum _{p\ge 0}\sum _{A\in \mathring I_E^p}(-1)^p\) is the Euler characteristic of the Čech complex \(C_{\mathring E}=\{\prod _{A\in \mathring I^p_E}C(U_A)\}_{p\ge 0}\), where \(C(U)\) denotes the module of constant functions on \(U\). Let \(F=E\setminus \mathring E\), which is also an extremal subset. Consider the Čech complexes \(C_E=\{\prod _{A\in I^p_E}C(U_A)\}_{p\ge 0}\) and \(C_F=\{\prod _{A\in I^p_F}C(U_A)\}_{p\ge 0}\). Note that \(C_{\mathring E}\) is the quotient of \(C_E\) by \(C_F\). Recall that \(\{U_\alpha \cap E\}_{\alpha \in I_E}\) and \(\{U_\alpha \cap F\}_{\alpha \in I_F}\) are topological good covers of \(E\) and \(F\), respectively. Hence the cohomologies of \(C_E\) and \(C_F\) are isomorphic to the singular cohomology groups \(H^\ast (E)\) and \(H^\ast (F)\), respectively, via spectral sequences. Moreover, since the contractions of the good covers are given by the same gradient flows, these natural isomorphisms commute with the horizontal exact sequences in the following diagram:

\[ \begin {CD} @>>> H^*(C_{\mathring E}) @>>> H^*(C_E) @>>> H^*(C_F) @>>> \\ @. @VVV @VVV @VVV \\ @>>> H^*(E,F) @>>> H^*(E) @>>> H^*(F) @>>> . \end {CD} \]

Therefore \(H^*(C_{\mathring E})\) is isomorphic to \(H^\ast (E,F)\) (see also the following Remark 15 ). By Proposition 13 , \(H^\ast (E,F)\) is isomorphic to the cohomology group \(H^\ast _c(\mathring E)\) with compact support. This completes the proof.

Remark 3.1.

More generally, the pair \((E,F)\) is homotopy equivalent to the geometric realization of the pair of the nerves of the restricted good covers \((\{U_\alpha \cap E\}_{\alpha \in I_E},\{U_\alpha \cap F\}_{\alpha \in I_F})\). This also yields the above isomorphism. See [13], Complement 8.4 for the proof.

Remark 3.2.

As shown in the above proof, the Betti numbers with compact support of \(\mathring E\) are finite. Moreover, they are uniformly bounded in terms of dimension, a lower curvature bound, and an upper diameter bound, since the Betti numbers of any extremal subset are uniformly bounded in terms of these constants ( [10], Theorem 1.1(2) ).

Remark 3.3.

The Betti numbers without compact support of \(\mathring E\) are also uniformly bounded. Indeed, since the distance function from \(F\) is regular near it ( [23], Lemma 3.1(2) ), the complement of a small metric neighborhood of \(F\) in \(E\) is a deformation retract of \(\mathring E\) by the gradient flow. This, together with the proof of the uniform boundedness in [10], Section 5 (cf. [28], Theorem 5.2 ), implies the claim.

Remark 3.4.

In view of Remark 14 , if \(G\) be an extremal subset of \(M\), then

\[\chi (G)=\sum _{E\in \mathcal E}\chi _c(\mathring E)\cdot \chi (F_E\cap G),\]

where we regard \(\chi (F_E\cap G)=0\) if \(F_E\cap G=\emptyset \). The proof is the same as above.

In this appendix, to complete the proof of Alesker's conjecture, we construct the integer valued function \(F\) on the strata of the limit space asserted in Conjecture 2 (2b). This is an unpublished result of Petrunin stated in [1], Theorem 4.6 without proof. As before, we fix and omit the coefficient field of singular cohomology.

Proof.