For a geometry $X$ (such as Euclidean, spherical, or hyperbolic) with isometry group $G$ the scissors congruence group $\mathcal{P}(X, G)$ is defined to be the free abelian group generated by polytopes in $X$, modulo the relation that for polytopes $P$ and $Q$ that intersect only on the boundary, $[P \cup Q]=[P]+[Q]$, and for $g \in G,[P]=[g \cdot P]$. This group classifies polytopes up to 'scissors congruence', i.e. cutting up into pieces, rearranging the pieces, and gluing them back together. With some basic group homology one can see that $\mathcal{P}(X, G) \cong H_0(G, \mathcal{P}(X, 1))$. Using combinatorial $K$-theory $\mathcal{P}(X, G)$ can be expressed as the $K_0$ of a spectrum $K(X, G)$. In this talk we will generalize this formula to show that, in fact, $K(X, G) \simeq K(X, 1)_{h G}$, and in fact more generally that this is true for any assembler with a $G$-action.This is joint work with Anna Marie Bohmann, Teena Gerhardt, Cary Malkiewich, and Mona Merling.[-]

In a seminal paper Henry Whitehead introduced four elementary operators, collapses and expansions (the inverse of a collapse), perforations and fillings (the inverse of a perforation), which correspond to an homotopy equivalence between two simplicial complexes. In this talk, we consider some transformations which are obtained by the means of these four operators. The presentation is composed of two parts. We begin the first part by introducing a certain axiomatic approach for combinatorial topology, which is settled in the framework of completions. Completions are inductive properties which may be expressed in a declarative way and may be combined. Then, we present a transformation that is based solely on collapses and expansions. This transformation involves homotopic pairs, it may be seen as a refinement of simple homotopy, which takes as input a single object. A homotopic pair is a couple of objects (X, Y ) such that X is included in Y and (X, Y ) may be transformed to a trivial couple by collapses and expansions that keep X inside Y . Our main result states that the collection of all homotopic pairs may be fully described by four completions which correspond to four global properties. After, we consider a transformation that is based on collapses, expansions, perforations, and fillings. This transformation involves contractible pairs, which are extensions of homotopic pairs. Again we show that the collection of all contractible pairs may be fully described by four completions which correspond to four global properties. Three of these completions are the same as the ones describing homotopic pairs. In the second part of the presentation, we introduce the notion of a Morse sequence, which provides a very simple approach to discrete Morse theory. A Morse sequence is obtained by considering only expansions and fillings of a simplicial complex, or, in a dual manner, by considering only collapses and perforations. A Morse sequence may be seen as an alternative way to represent the gradient vector field of an arbitrary discrete Morse function. We introduce reference maps, which are maps that associate a set of critical simplexes to each simplex appearing in a Morse sequence. By considering the boundary of each critical simplex, we obtain a chain complex from these maps, which corresponds precisely to the Morse complex. Then, we define extension maps. We show that, when restricted to homology, an extension map is the inverse of a reference map. Also we show that these two maps allow us to recover directly the isomorphism theorem between the homology of an object and the homology of its Morse complex[-]