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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.