Geometric modules and algebraicK-homology theory

Anderson, Douglas R.; Munkholm, Hans J.

doi:10.1007/bf01054451

Cited by 16 publications

(6 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is precisely the metric used by Anderson and Munkholm in [AM90] and also by Siebenmann and Sullivan in [SS79], but there is a notable distinction: we do not necessarily require that our metric space (M, d) have a finite bound. Define the coning map j X : X × R → O(X + ) as the natural quotient map…”

Section: Is a Bounded Homotopy Equivalence Measured In The Open Conementioning

confidence: 99%

Algebraic subdivision in simplicially controlled categories

Adams-Florou

2014

Preprint

View full text Add to dashboard Cite

We generalise the notion of subdivision of a finite-dimensional locally finite simplicial complex X to geometric algebra, namely to the simplicially controlled categories A * (X), A * (X) of Ranicki and Weiss. We prove a squeezing result: a bounded chain equivalence of sufficiently algebraically subdivided chain complexes can be squeezed to a simplicially controlled chain equivalence of the unsubdivided chain complexes. Giving X × R a bounded triangulation measured in the open cone O(X + ) we use algebraic subdivision to define a functor " − ⊗Z ′′ : B(A(X)) → B(A(X × R)) that corresponds to tensoring with the simplicial chain complex of Z and algebraically subdividing to be bounded over O(X + ). We show that C ≃ 0 ∈ B(A(X)) if and only if "C ⊗ Z ′′ is boundedly chain contractible over O(X + ). These results have applications to Poincaré duality and homology manifold detection as a finite-dimensional locally finite simplicial complex X is a homology manifold if and only if it has X-controlled Poincaré duality. We prove a Poincaré duality squeezing theorem that such a space X with sufficiently controlled Poincaré duality must have X-controlled Poincaré duality and we prove a Poincaré duality splitting theorem with the consequence that X is a homology manifold if and only if X × R has bounded Poincaré duality over O(X + ).

show abstract

Section: Is a Bounded Homotopy Equivalence Measured In The Open Conementioning

confidence: 99%

Algebraic subdivision in simplicially controlled categories

Adams-Florou

2014

Preprint

View full text Add to dashboard Cite

show abstract

“…The open cone was first considered by Pedersen and Weibel in [PW89] where it was defined for subsets of S n . This definition was extended to more general spaces by Anderson and Munkholm in [AM90]. We make the following definition: For a complete metric space…”

Section: Preliminariesmentioning

confidence: 99%

A controlled local-global theorem for simplicial complexes

Adams-Florou

2013

Preprint

View full text Add to dashboard Cite

In this paper we prove that a simplicial map of finite-dimensional locally finite simplicial complexes has contractible point inverses if and only if it is an ǫcontrolled homotopy equivalence for all ǫ > 0 if and only if f × id R is a bounded homotopy equivalence measured in the open cone over the target. This confirms for such a space X the slogan that arbitrarily fine control over X corresponds to bounded control over the open cone O(X+). For the proof a one parameter family of cellulations {X ′ ǫ } 0<ǫ<ǫ(X) is constructed which provides a retracting map for X which can be used to compensate for sufficiently small control.

show abstract

“…(O(B); O(p)) be the category of boundedly controlled geometric modules defined in Section 3. The reader will see that this construction is slightly different from the open cone construction used in [AM2] in that it puts a copy of X, rather than just a single point, over the cone point in O(B). Nonetheless, the arguments used in [AM2], combined with those of this paper, show that the collection of spaces K bc (ξ) = {K GM bc (S n ξ)} may be endowed with the structure of an Ω-spectrum and that the functor K bc : T OP/CM * → SPEC is a homology theory on T OP/CM * .…”

Section: Some Calculationsmentioning

confidence: 99%

“…It is also possible to use the boundedly controlled geometric module theory of [AM2] to define a functor GM bc : T OP/CM * → ADDCAT .…”

mentioning

confidence: 99%

See 1 more Smart Citation

Continuously controlled K-theory with variable coefficients

Munkholm

Anderson

2000

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

A continuing theme in "controlled topology" over the last fifteen years has been to use geometric modules to define and investigate algebraic invariants of geometric problems. One impetus to this approach was the work of Quinn [Qn1], [Qn2], [Qn3], and [Qn4] although the idea of a geometric module preceeded it by several years. Subsequently, many authors have studied different forms of geometric module theory and explored their properties. Among these are Pedersen [P] and Pedersen and Weibel [PW1] and [PW2], who studied the additive categories of what we now call geometric modules with bounded control and constant coefficients and showed how to use their algebraic K-theory to construct a homology theory on the category of compact Lipschitz spaces. The present authors [AM2] developed these ideas further by using geometric modules with bounded control and variable coefficients to construct a homology theory on a certain category of maps T OP/LIP c . More recently Anderson, Connolly, Ferry, and Pedersen [ACFP] introduced the notion of geometric modules with continuous control at infinity and used the constant coefficient case of this theory to construct a homology theory on the category of compact metrizable spaces. This point of view was subsequently used in [AC] and [CP].Although it has been clear for some time that there should be a "pushout" of these three approaches in which a homology theory is constructed by using geometric modules with continuous control at infinity and variable coefficients, it has been less clear whether such a theory would have any new applications and, at a more technical level, what the domain of such a theory should be. The former question, however, is resolved in [ACM]. That paper completes a series of papers by the present authors [AM2] and [AM3] that compare their approach to K-theory in [AM1] to the approaches of other researchers by examining the relationship between their work and the "controlled" K-theory developed by Quinn [Qn3] and [Qn4], Yamasaki [Y] and Ranicki and Yamasaki [RY]. The K-theory defined and developed in this paper plays a central and natural role in [ACM].In this paper we introduce the category T OP cc /LC in which an object, (X, B; p), is a map p : E → X where X = X − B, X is a locally compact, Hausdorff space, and B is a compact subspace. We then define a functor GM cc : T OP cc /LC → ADDCAT , the category of additive categories. It assigns to the object (X, B; p) in T OP cc /LC, the category GM cc (X, B; p) whose objects are geometric modules in E and whose morphisms are path matrix morphisms having continuous control at B.

show abstract

Geometric modules and algebraicK-homology theory

Cited by 16 publications

References 20 publications

Algebraic subdivision in simplicially controlled categories

Algebraic subdivision in simplicially controlled categories

A controlled local-global theorem for simplicial complexes

Continuously controlled K-theory with variable coefficients

Contact Info

Product

Resources

About