Department of Mathematics Mathematics 416, section X1 Topics in Algebra, Algebraic K-theory 12-12:50 PM MWF 343 Altgeld Hall Professor Daniel R. Grayson Email: <dan@math.uiuc.edu> Office: 357 Altgeld Hall |

**Introduction:**

Algebraic K-theory is the application of homotopy theory to problems in linear algebra arising from diverse areas such as number theory and algebraic geometry.

The spaces whose homotopy groups are considered are constructed by totally combinatorial means. For example, we may start with the category of finite dimensional vectors spaces over a field K, and use it to construct various other categories. From one of those categories we may make a topological space by gluing line segments, triangles, tetrahedra, and so on: each object of the category would correspond to a point of the space, each arrow to a line segment joining two previously deposited points, each commutative triangle of arrows to triangle glued to three previously deposited intervals, and so on. We develop of dictionary between category theory and homotopy theory, according to which functors become continuous maps, natural transformations become homotopies, equivalences of categories become homotopy equivalences of spaces, and categories with a final object become contractible spaces. The final cornerstone of the dictionary is this: some triples of spaces called fibration sequences yield long exact sequences of homotopy groups; Quillen's Theorem B provides a criterion for a triple of categories to become a fibration sequence. We will cover the application of this dictionary to develop results about higher K-theory.

**The textbook:**

Copies of relevant papers will be distributed. The book "Introduction to Algebraic K-Theory", by John Willard Milnor, Princeton University Press, 1972, ISBN: 0-691-08101-8, is recommended, but not required, as a good introduction to lower algebraic K-theory, and thus can serve as background for this course, which will concentrate on higher algebraic K-theory.

**Useful previous courses:**

I expect every student to have had Math 402 and some basic point set topology. If we need a theorem from a more advanced course we'll either prove it or at least state it carefully. Other math courses that would serve as useful preparation would include any from the following list: 430, 431, 432, 422, 406, 407.

**Course notes, frozen April 30, 2003:**

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