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2. $K$-Theory and Algebraic Topology


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      Problem 2.1.

      Is there a direct scanning map proof that scissors congruence $K$-theory of polytopes is a Thom spectrum?


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          Problem 2.2.

          Is there a $K$-theory spectrum that has homology groups $H_i(SL_n(\Z);St(\Q))$?


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              Problem 2.3.

              Develop projectivised scissors congruence $K$-theory. (For instance, can you prove polytopes with only vertices on the quadric generate the group?)


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                  Problem 2.4.

                  Develop universal characterisation for $K$-theory for the following kinds of categories:

                  • 1) Monoidal Categories
                  • 2) Exact Categories, CGW categories, Subtractive Categories
                  • 3) Multicategories
                  • 4) Squares


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                      Problem 2.5.

                      What is the relationship between total scissors congruence & discrete orthogonal $K$-Theory? Role of rank filtration of the associated $S_\bullet$-construction?

                          Cite this as: AimPL: Scissors congruences, algebraic K-theory and Steinberg modules, available at http://aimpl.org/scissorssteinberg.