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\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

2. A-homotopy theory

    1. Homotopy colimits in A-homotopy theory


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      Conjecture 2.1.

      [Daniel Carranza, Chris Kapulkin] The category of graphs with weak equivalences given by the maps inducing isomorphisms on all A-homotopy groups admits (small) homotopy colimits.
          This statement is wide open. There are other questions that are special cases that are also unknown. We give a simple example below.

      For a graph X, define the n-suspension of X to be the graph \Sigma_n X = X \square I_n/\sim, where \sim identifies: (x, 0) \sim (x', 0) and (x, n) \sim (x', n) for all x, x' \in X. For n \geq 5, are \Sigma_{n+1} X and \Sigma_n X weak homotopy equivalent? This is not known even for n = 5 and X = C_5.

      Eric Babson suggested asking if the map \Sigma_{n+1} X \rightarrow \Sigma_n X collapsing a level is m-connected (i.e., induces an isomorphism on A_k’s for k \leq m) and allow n to depend on n.

        • Applications to subspace arrangements


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          Problem 2.2.

          Can earlier results of Barcelo on the relation between A-theory and subspace arrangements be extended beyond A_1 by showing that the higher A-groups vanish?

            • Computation of cubical homology

                  When computing cubical homology of a simple graph X, one builds a cubical set MX whose n-cubes are maps I_1^{\square n} \to X and then takes the usual cubical homology of this cubical set.

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              Problem 2.3.

              [Samira Jamil] Can cubical homology of a graph be computed using only injective maps I_1^{\square n} \to X? Should this restriction be made at the cubical set or chain complex level?
                  A similar question in the context of digital topology was considered by Jamil in her Ph.D. dissertation.

                • Classifying spaces


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                  Problem 2.4.

                  [Eric Babson] Do there exist classifying graphs for fibrations with a fixed fiber (up to weak equivalence) in A-theory? Here, by fibrations we mean maps satisfying the graph analogue of the sharp map condition from simplicial sets.

                      Cite this as: AimPL: Discrete and combinatorial homotopy theory, available at http://aimpl.org/combhomotop.