Many important spaces in topology and algebraic geometry have no odd-dimensional homology. For such spaces, functorial spatial homology truncation simplifies considerably. On the theory side, the simplification arises as follows: To define general spatial homology truncation, we used intermediate auxiliary structures, the n-truncation structures. For spaces that lack odd-dimensional homology, these structures can be replaced by a much simpler structure. Again every such space can be embedded in such a structure, which is the analogon of the general theory. On the application side, the crucial simplification is that the truncation functor t_{<n} will not require that in truncating a given continuous map, the map preserve additional structure on the domain and codomain of the map. In general, t_{<n} is defined on the category CW_{n⊃∂}, meaning that a map must preserve chosen subgroups “Y ”. Such a condition is generally necessary on maps, for otherwise no truncation exists. So arbitrary continuous maps between spaces with trivial odd-dimensional homology can be functorially truncated. In particular the compression rigidity obstructions arising in the general theory will not arise for maps between such spaces.

Let ICW be the full subcategory of CW whose objects are simply connected CW-complexes K with finitely generated even-dimensional homology and vanishing odd-dimensional homology for any coefficient group. We call ICW the interleaf category.

For example, the space K = S^{2} ∪_{2} e^{3} is simply connected and has vanishing integral homology in odd dimensions. However, H_{3}(K;Z/_{2}) = Z/_{2} ≠ 0.

Let X be a space whose odd-dimensional homology vanishes for any coefficient group. Then the even-dimensional integral homology of X is torsion-free.

Taking the coefficient group Q/Z, we have

Tor(H_{2k}(X),Q/Z) = H_{2k+1}(X) ⊗ Q/Z ⊕ Tor(H_{2k}(X),Q/Z) = H_{2k+1}(X;Q/Z) = 0.

Thus H_{2k}(X) is torsion-free, since the group Tor(H_{2k}(X),Q/Z) is isomorphic to the torsion subgroup of H_{2k}(X).

Any simply connected closed 4-manifold is in ICW. Indeed, such a manifold is homotopy equivalent to a CW-complex of the form

V_{i=1}^{k}S_{i}^{2}∪_{ƒ}e^{4}

where the homotopy class of the attaching map ƒ : S^{3} → V_{i=1}^{k} S_{i}^{2} may be viewed as a symmetric k × k matrix with integer entries, as π_{3}(V_{i=1}^{k}S_{i}^{2}) ≅ M(k), with M(k) the additive group of such matrices.

Any simply connected closed 6-manifold with vanishing integral middle homology group is in ICW. If G is any coefficient group, then H_{1}(M;G) ≅ H_{1}(M) ⊗ G ⊕ Tor(H_{0}M,G) = 0, since H_{0}(M) = Z. By * Poincaré duality*,

0 = H_{3}(M) ≅ H_{3}(M) ≅ Hom(H_{3}M,Z) ⊕ Ext(H_{2}M,Z),

so that H_{2}(M) is free. This implies that Tor(H_{2}M,G) = 0 and hence H_{3}(M;G) ≅ H_{3}(M) ⊗ G ⊕ Tor(H_{2}M,G) = 0. Finally, by G-coefficient Poincaré duality,

H_{5}(M;G) ≅ H_{1}(M;G) ≅ Hom(H_{1}M,G) ⊕ Ext(H_{0}M,G) = Ext(Z,G) = 0

Any smooth, compact toric variety X is in ICW: * Danilov’s Theorem* implies that H

^{∗}(X;Z) is torsion-free and the map A

^{∗}(X) → H

^{∗}(X;Z) given by composing the canonical map from

*to homology, A*

**Chow groups**^{k}(X) = A

_{n−k}(X) → H

_{2n−2k}(X;Z), where n is the complex dimension of X, with Poincaré duality H

_{2n−2k}(X;Z) ≅ H

_{2k}(X;Z), is an isomorphism. Since the odd-dimensional cohomology of X is not in the image of this map, this asserts in particular that H

_{odd}(X;Z) = 0. By Poincaré duality, H

_{even}(X;Z) is free and H

_{odd}(X;Z) = 0. These two statements allow us to deduce from the universal coefficient theorem that H

_{odd}(X;G) = 0 for any coefficient group G. If we only wanted to establish H

_{odd}(X;Z) = 0, then it would of course have been enough to know that the canonical, degree-doubling map A

^{∗}(X) → H

^{∗}(X;Z) is onto. One may then immediately reduce to the case of projective toric varieties because every complete fan Δ has a projective subdivision Δ

^{′}, the corresponding proper

*X(Δ*

**birational morphism**^{′}) → X(Δ) induces a surjection H

^{∗}(X(Δ

^{′});Z) → H

^{∗}(X(Δ);Z) and the diagram

commutes.

Let G be a complex, simply connected, * semisimple Lie group* and P ⊂ G a connected parabolic subgroup. Then the homogeneous space G/P is in ICW. It is simply connected, since the fibration P → G → G/P induces an exact sequence

1 = π_{1}(G) → π_{1}(G/P) → π_{0}(P) → π_{0}(G) = 0,

which shows that π_{1}(G/P) → π_{0}(P) is a bijection. Accordingly, ∃ elements s_{w}(P) ∈ H_{2l(w)}(G/P;Z) (“* Schubert classes*,” given geometrically by

*), indexed by w ranging over a certain subset of the*

**Schubert cells***of G, that form a basis for H*

**Weyl group**_{∗}(G/P;Z). (For w in the Weyl group, l(w) denotes the length of w when written as a reduced word in certain specified generators of the Weyl group.) In particular H

_{even}(G/P;Z) is free and H

_{odd}(G/P;Z) = 0. Thus H

_{odd}(G/P;G) = 0 for any coefficient group G.

The linear groups SL(n, C), n ≥ 2, and the subgroups S p(2n, C) ⊂ SL(2n, C) of transformations preserving the alternating bilinear form

x_{1}y_{n+1} +···+ x_{n}y_{2n −xn+1y1 −···−x2nyn}

on C^{2n} × C^{2n} are examples of complex, simply connected, semisimple Lie groups. A parabolic subgroup is a closed subgroup that contains a * Borel group* B. For G = SL(n,C), B is the group of all upper-triangular matrices in SL(n,C). In this case, G/B is the complete flag manifold

G/B = {0 ⊂ V_{1} ⊂···⊂ V_{n−1} ⊂ C_{n}}

of flags of subspaces V_{i} with dimV_{i} = i. For G = Sp(2n,C), the Borel subgroups B are the subgroups preserving a half-flag of isotropic subspaces and the quotient G/B is the variety of all such flags. Any parabolic subgroup P may be described as the subgroup that preserves some partial flag. Thus (partial) flag manifolds are in ICW. A special case is that of a maximal parabolic subgroup, preserving a single subspace V. The corresponding quotient SL(n, C)/P is a * Grassmannian* G(k, n) of k-dimensional subspaces of C

^{n}. For G = Sp(2n,C), one obtains Lagrangian Grassmannians of isotropic k-dimensional subspaces, 1 ≤ k ≤ n. So Grassmannians are objects in ICW. The interleaf category is closed under forming

*.*

**fibrations**