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There is a construction of the algebraic K-theory groups $K_i(R)$ of a ring $R$ by Volodin. He gave an explicit construction of the plus-construction $BGL(R)^+$ as the quotient of the bar construction $BGL(R)$ by the union $\bigcup_{n,\sigma} BU_n(R)^\sigma$ of the classifying space of the group $U_n(R)$ upper-triangular matrices with ones on the diagonal, conjugated by a permutation matrix $\sigma$.

What are the uses of this construction in modern approaches to algebraic K-theory? What are its advantages and disadvantages?

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It has several uses:

  1. Volodin $K$-theory was used by Igusa in the late 1970s/Early 1980s to define $K$-theoretic invariants of families of pseudoisotopies.

  2. In the 1980s, the Volodin construction (actually a variant of it) was used by Goodwillie to relate rational relative K-theory to rational relative THH. This was further developed by Dundas and McCarthy in their famous result that stable K-theory is isomorphic to THH. This is the genesis of trace methods in algebraic K-theory.

  3. In the early 1990s, Suslin and Wodzicki used Volodin's space to obtain excision results for algebraic $K$-theory.

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