Finiteness Obstruction Research Articles

Cancellation for (G,n)-complexes and the Swan Finiteness Obstruction

Abstract In previous work, we related homotopy types of finite $(G,n)$-complexes when $G$ has periodic cohomology to projective ${\mathbb{Z}} G$-modules representing the Swan finiteness obstruction. We use this to determine when $X \vee S^{n} \simeq Y \vee S^{n}$ implies $X \simeq Y$ for finite $(G,n)$-complexes $X$ and $Y$, and give lower bounds on the number of homotopically distinct pairs when this fails. The proof involves constructing projective ${\mathbb{Z}} G$-modules as lifts of locally free modules over orders in products of quaternion algebras, whose existence follows from the Eichler mass formula. In the case $n=2$, difficulties arise that lead to a new approach to finding a counterexample to Wall’s D2 problem.

Hamiltonicity of claw-free graphs and Fan-type conditions

For a graph H, let δF(H)=min{max{d(u),d(v)}|for any u,v∈V(H) with distH(u,v)=2}. For a given integer p>0 and a given number ϵ, a graph H of order n is said to satisfy a Fan-type condition if δF(H)≥2n+ϵp. In this paper, we show that like the planar graphs having obstruction set {K5,K3,3}, for the family of k-connected claw-free Hamiltonian graphs H with δF(H)≥2n+ϵp, if k=3 it has a finite obstruction set in which each graph has order at most max{9p∕2−20,10}, and if k=2 it has an obstruction set in which each graph has at most max{9p−20,0} vertices of degree greater than 3. We determine the best possible values of p and ϵ when the obstruction set is empty. We show that for k-connected claw-free simple graphs H of order n with n large enough, each of the following holds:(a) if k=2 and δF(H)≥2n−114 then H is Hamiltonian;(b) if k=3 and δF(H)≥2n−198 then H is Hamiltonian.These bounds are sharp. For k=3 and a given integer p, since the obstruction set is finite and can be determined in polynomial time, improvements to (b) for other given values of p are possible with the help of a computer.

Wall's finiteness obstruction

A short complete description of Wall's finiteness obstruction.

Diagonal resolutions for metacyclic groups

We show the finite metacyclic groups G(p,q) admit a class of projective resolutions which are periodic of period 2q and which in addition possess the properties that a) the differentials are 2×2diagonal matrices; b) the Swan–Wall finiteness obstruction (cf. [19,20]) vanishes. We obtain thereby a purely algebraic proof of Petrie's Theorem ([14]) that G(p,q) has free period 2q.

Excluded vertex-minors for graphs of linear rank-width at most [formula omitted

Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each k, there is a finite obstruction set Ok of graphs such that a graph G has linear rank-width at most k if and only if no vertex-minor of G is isomorphic to a graph in Ok. However, no attempts have been made to bound the number of graphs in Ok for k≥2. We show that for each k, there are at least 2Ω(3k) pairwise locally non-equivalent graphs in Ok, and therefore the number of graphs in Ok is at least double exponential.To prove this theorem, it is necessary to characterize when two graphs in Ok are locally equivalent. A graph is a block graph if all of its blocks are complete graphs. We prove that if two block graphs without simplicial vertices of degree at least 2 are locally equivalent, then they are isomorphic. This not only is useful for our theorem but also implies a theorem of Bouchet (1988) stating that if two trees are locally equivalent, then they are isomorphic.

Euler characteristics of categories and homotopy colimits

In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L^2 -Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of \mathcal I -indexed categories where \mathcal I is any small category admitting a finite \mathcal I - CW -model for its \mathcal I -classifying space. Special cases of our Homotopy Colimit Formula include formulas for products, homotopy pushouts, homotopy orbits, and transport groupoids. We also apply our formulas to Haefliger complexes of groups, which extend Bass–Serre graphs of groups to higher dimensions. In particular, we obtain necessary conditions for developability of a finite complex of groups from an action of a finite group on a finite category without loops.

Finiteness obstructions and Euler characteristics of categories

We introduce notions of finiteness obstruction, Euler characteristic, L 2 -Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type ( FP R ) is a class in the projective class group K 0 ( R Γ ) ; the functorial Euler characteristic and functorial L 2 -Euler characteristic are respectively its RΓ-rank and L 2 -rank. We also extend the second author's K-theoretic Möbius inversion from finite categories to quasi-finite categories. Our main example is the proper orbit category, for which these invariants are established notions in the geometry and topology of classifying spaces for proper group actions. Baez and Dolan's groupoid cardinality and Leinster's Euler characteristic are special cases of the L 2 -Euler characteristic. Some of Leinster's results on Möbius–Rota inversion are special cases of the K-theoretic Möbius inversion.

A relative version of the finiteness obstruction theory of C. T. C. Wall

In his 1965 paper C. T. C. Wall demonstrated that if a CW complex Y is finitely dominated, then the reduced projective class group of Y contains an obstruction which vanishes if and only if Y is homotopy equivalent to a finite CW complex. Wall also demonstrated that such an obstruction is invariant under homotopy equivalences. Subsequently Sum and Product Theorems for this obstruction were proved by L. C. Siebenmann. In his second paper on the subject Wall gives an algebraic definition of the relative finiteness obstruction. If a CW complex Y is finitely dominated rel. a subcomplex X, then the reduced projective class group of Y contains an obstruction which vanishes if and only if Y is homotopy equivalent to a finite complex rel. X. In this paper we will use a geometric construction to reduce the relative finiteness obstruction to the non-relative version. We will demonstrate that the relative finiteness obstruction is invariant under certain types of homotopy equivalences. We will also prove the relative versions of the Sum and the Product Theorems.

Achievable sets, brambles, and sparse treewidth obstructions

One consequence of the graph minor theorem is that for every k there exists a finite obstruction set Obs ( TW ⩽ k ) . However, relatively little is known about these sets, and very few general obstructions are known. The ones that are known are the cliques, and graphs which are formed by removing a few edges from a clique. This paper gives several general constructions of minimal forbidden minors which are sparse in the sense that the ratio of the treewidth to the number of vertices n does not approach 1 as n approaches infinity. We accomplish this by a novel combination of using brambles to provide lower bounds and achievable sets to demonstrate upper bounds. Additionally, we determine the exact treewidth of other basic graph constructions which are not minimal forbidden minors.

Nonstabilized Nielsen coincidence invariants and Hopf–Ganea homomorphisms

In classical fixed point and coincidence theory the notion of Nielsen numbers has proved to be extremely fruitful. We extend it to pairs (f_1,f_2) of maps between manifolds of arbitrary dimensions, using nonstabilized normal bordism theory as our main tool. This leads to estimates of the minimum numbers MCC(f_1,f_2) (and MC(f_1,f_2), respectively) of path components (and of points, resp.) in the coincidence sets of those pairs of maps which are homotopic to (f_1,f_2). Furthermore, we deduce finiteness conditions for MC(f_1,f_2). As an application we compute both minimum numbers explicitly in various concrete geometric sample situations. The Nielsen decomposition of a coincidence set is induced by the decomposition of a certain path space E(f_1,f_2) into path components. Its higher dimensional topology captures further crucial geometric coincidence data. In the setting of homotopy groups the resulting invariants are closely related to certain Hopf--Ganea homomorphisms which turn out to yield finiteness obstructions for MC.

The Euler Characteristic and Finiteness Obstruction of Manifolds with Periodic Ends

Let M be a complete orientable manifold of bounded geometry. Suppose that M has finitely many ends, each having a neighborhood quasi-isometric to a neighborhood of an end of an infinite cyclic covering of a compact manifold. We consider a class of exponentially weighted inner products (·, ·) k on forms, indexed by k > 0. Let δk be the formal adjoint of d for (·, ·) k . It is shown that if M has finitely generated rational homology, d+δk is Fredholm on the weighted spaces for all sufficiently large k. The index of its restriction to even forms is the Euler characteristic of M. This result is generalized as follows. Let π = π1 (M) . Take d+ δk with coefficients in the canonical C∗ (π)-bundle ψ overM. If the chains of M with coefficients in ψ are C∗ (π)-finitely dominated, then d+δk is Fredholm in the sense of Mĭscenko and Fomenko for all sufficiently large k. The index in K0 (C ∗ (π)) is related to Wall’s finiteness obstruction. Examples are given where it is nonzero.

Homotopy and homology of fibred spaces

We study fibred spaces with fibres in a structure category F and we show that cellular approximation, Blakers–Massey theorem, Whitehead theorems, obstruction theory, Hurewicz homomorphism, Wall finiteness obstruction, and Whitehead torsion theorem hold for fibred spaces. For this we introduce the cohomology of fibred spaces.

Manifolds with non-stable fundamental groups at infinity, II

In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann's famous collaring theorem that may be applied to manifolds having non-stable fundamental group systems at infinity. In this paper we show that, for manifolds with compact boundary, the condition of inward tameness has substatial implications for the algebraic topology at infinity. In particular, every inward tame manifold with compact boundary has stable homology (in all dimensions) and semistable fundamental group at each of its ends. In contrast, we also construct examples of this sort which fail to have perfectly semistable fundamental group at infinity. In doing so, we exhibit the first known examples of open manifolds that are inward tame and have vanishing Wall finiteness obstruction at infinity, but are not pseudo-collarable.

The structure of obstructions to treewidth and pathwidth

It is known that the class of graphs with treewidth (resp. pathwidth) bounded by a constant w can be characterized by a finite obstruction set obs( TW( w)) (resp. obs( PW( w))). These obstruction sets are known for w⩽3 so far. In this paper we give a structural characterization of graphs from obs( TW( w)) (resp. obs( PW( w))) with a fixed number of vertices in terms of subgraphs of the complement. Our approach also essentially simplifies known characterization of graphs from obs( TW( w)) (resp. obs( PW( w))) with ( w+3) vertices. Also for any w⩾3 a graph from obs( TW( w))⧹ obs( PW( w)) is constructed, that solves an open problem.

Forbidden minors to graphs with small feedback sets

Finite obstruction set characterizations for lower ideals in the minor order are guaranteed to exist by the graph minor theorem. In this paper we characterize several families of graphs with small feedback sets, namely k 1-F EEDBACK V ERTEX S ET, k 2-F eedback E DGE S ET and ( k 1, k 2)-F EEDBACK V ERTEX/E DGE S ET, for small integer parameters k 1 and k 2. Our constructive methods can compute obstruction sets for any minor-closed family of graphs, provided the pathwidth (or treewidth) of the largest obstruction is known.

The finiteness obstruction for loop spaces

For finitely dominated spaces, Wall constructed a finiteness obstruction, which decides whether a space is equivalent to a finite CW-complex or not. It was conjectured that this finiteness obstruction always vanishes for quasi finite H-spaces, that are H-spaces whose homology looks like the homology of a finite CW-complex. In this paper we prove this conjecture for loop spaces. In particular, this shows that every quasi finite loop space is actually homotopy equivalent to a finite CW-complex.

The structure of obstructions to treewidth and pathwidth

It is known that the class of graphs with treewidth (resp. pathwidth) bounded by a constant w can be characterized by a finite obstruction set obs(TW(w)) (resp. obs(PW(w))). These obstruction sets are known for w ≤ 3 so far. In this paper we give a structural characterization of graphs from obs(TW(w)) (resp. obs(PW(w))) with a fixed number of vertices in terms of subgraphs of the complement. Our approach also essentially simplifies known characterization of graphs from obs(TW(w)) (resp. obs(PW(w))) with (w + 3) vertices. Also for any w ≥ 3 a graph from obs(TW(w))\obs(PW(w)) is constructed, that solves an open problem.

Existence of finitely dominatedCW-complexes withG 1(X) = π1(X) and non-vanishing finiteness obstruction

We show for a finite abelian groupG and any element in the image of the Swan homomorphism sw:\(\mathbb{Z}/|G|* \to \tilde K_0 (\mathbb{Z}G)\) that it can be realized as the finiteness obstruction of a finitely dominated connectedCW-complexX with fundamental group π1(X) =G such that π1(X) is equal to the subgroupG1(X) defined by Gottlieb. This is motivated by the observation that anyH-spaceX satisfies π1(X) =G1(X) and still the problem is open whether any finitely dominatedH-space is up to homotopy finite.

Controlled K-theory

The controlled finiteness obstruction and torsion are defined using controlled algebra, giving a more algebraic proof of the topological invariance of torsion and the homotopy finiteness of compact ANRs.

Finiteness obstructions and cocompact actions onS m×ℝn

Finiteness obstructions and cocompact actions onS m×ℝn