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## Complexity and Good Spaces

### Jeff Strom

Western Michigan University

###
Thursday, April 28, 2005

L02 Carson Hall, 4 pm

Tea 3:30 pm, Math Lounge

**Abstract: ** In algebraic topology, we like to build new
spaces out of ones we already understand. One procedure for doing
this is called the homotopy colimit. For any space *A*, we can form
the smallest collection of spaces which contains *A* and which is
closed under the operation of taking homotopy colimits. Such a
collection is called the __closed class__ generated by *A*, and
it is denoted *C(A)*.

Now if *X\in C(A)*, then
*X* can be obtained from *A* by repeating the homotopy colimit
operation, perhaps infinitely many times. Indeed, *X* could be
obtained in many different ways. The *A*-complexity of *X*
with respect to *A* is the minimum number of times you need to repeat
the homotopy colimit operation before getting *X*. It is denoted
*\kappa_A(X)*.\par It may happen that *C(A)* is, in addition
and by accident, closed under another construction----extensions by
fibrations. If this is the case, then we call *A* a __good
space__. Examples suggest that complexity with respect to good spaces
is relatively small. In this talk, I'll characterize good space and
use the characterization to prove that this intuition is correct
---complexity with respect to good spaces is (more or less) always
finite or (at worst) countably infinite.

This is joint work with
Michele Intermont. I will try to keep the really technical stuff
under the rug, so that the talk will be accessible to graduate
students who are comfortable with CW complexes.

This talk will be accessible to graduate students.