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Authors
James Gillespie
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We discuss some recent developments in the theory of abelian model categories. The emphasis is on the hereditary condition and applications to homotopy categories of chain complexes and stable module categories.
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Each object of any abelian model category has a canonical resolution as described in this article. When the model structure is hereditary we show how morphism sets in the associated homotopy category may be realized as cohomology groups computed from these resolutions. We also give an alternative description of the morphism sets in terms of Yoneda Ext groups.
Tate cohomology (as well as Borel homology and cohomology) of connective K-theory for $G=(mathbb{Z}/2)^n$ was completely calculated by Bruner and Greenlees. In this note, we essentially redo the calculation by a different, more elementary method, and we extend it to $p>2$ prime. We also identify the resulting spectra, which are products of Eilenberg-Mac Lane spectra, and finitely many finite Postnikov towers. For $p=2$, we also reconcile our answer completely with the result of Bruner and Greenlees, which is in a different form, and hence the comparison involves some non-trivial combinatorics.
We introduce the notion of hereditary G-compactness (with respect to interpretation). We provide a sufficient condition for a poset to not be hereditarily G-compact, which we use to show that any linear order is not hereditarily G-compact. Assuming that a long-standing conjecture about unstable NIP theories holds, this implies that an NIP theory is hereditarily G-compact if and only if it is stable (and by a result of Simon, this holds unconditionally for $aleph_0$-categorical theories). We show that if $G$ is definable over $A$ in a hereditarily G-compact theory, then $G^{00}_A=G^{000}_A$. We also include a brief survey of sufficient conditions for G-compactness, with particular focus on those which can be used to prove or disprove hereditary G-compactness for some (classes of) theories.
We investigate the parameterized complexity of finding subgraphs with hereditary properties on graphs belonging to a hereditary graph class. Given a graph $G$, a non-trivial hereditary property $Pi$ and an integer parameter $k$, the general problem $P(G,Pi,k)$ asks whether there exists $k$ vertices of $G$ that induce a subgraph satisfying property $Pi$. This problem, $P(G,Pi,k)$ has been proved to be NP-complete by Lewis and Yannakakis. The parameterized complexity of this problem is shown to be W[1]-complete by Khot and Raman, if $Pi$ includes all trivial graphs but not all complete graphs and vice versa; and is fixed-parameter tractable (FPT), otherwise. As the problem is W[1]-complete on general graphs when $Pi$ includes all trivial graphs but not all complete graphs and vice versa, it is natural to further investigate the problem on restricted graph classes. Motivated by this line of research, we study the problem on graphs which also belong to a hereditary graph class and establish a framework which settles the parameterized complexity of the problem for various hereditary graph classes. In particular, we show that: $P(G,Pi,k)$ is solvable in polynomial time when the graph $G$ is co-bipartite and $Pi$ is the property of being planar, bipartite or triangle-free (or vice-versa). $P(G,Pi,k)$ is FPT when the graph $G$ is planar, bipartite or triangle-free and $Pi$ is the property of being planar, bipartite or triangle-free, or graph $G$ is co-bipartite and $Pi$ is the property of being co-bipartite. $P(G,Pi,k)$ is W[1]-complete when the graph $G$ is $C_4$-free, $K_{1,4}$-free or a unit disk graph and $Pi$ is the property of being either planar or bipartite.
We study a categorical construction called the cobordism category, which associates to each Waldhausen category a simplicial category of cospans. We prove that this construction is homotopy equivalent to Waldhausens $S_{bullet}$-construction and therefore it defines a model for Waldhausen $K$-theory. As an example, we discuss this model for $A$-theory and show that the cobordism category of homotopy finite spaces has the homotopy type of Waldhausens $A(*)$. We also review the canonical map from the cobordism category of manifolds to $A$-theory from this viewpoint.
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