    High School Mathematics Contest Contest Home > Contest Logo -> 2005 Contest Logo

## 2005 Contest Logo - A covering Space   We owe the drawing of the T-shirt design and our 2005 contest logo to professor Larry Smolinsky from LSU Math Department. It portrays a covering space.

A covering map is a continous onto map

p : C X

with C and X being topological spaces, which has the following property:

to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open sets U~i (where i ranges over some index set I) such that p restricted to U~i yields a homeomorphism from U~i to U for every i in I.

We say C is a covering space of X.

Our 2005 contest logo portrays a complicated case of a covering space C. We will try to explain here a simpler example.   On the picture above C = R is potrayed as a spiral around a vertical cylinder, and X = S1 lies in a horizontal plane perpendicular to the axis of the cylinder.

Consider X being the unit circle S1 in R2.

Then the map

p : R S1

with

p(t) = (cos(t),sin(t))

is a covering map, and C = R is a covering space of X = S1.

In this case C = R can be drawn as a spiral: The picture on the right explains the details.  Contest organizer: Contest e-mail:   Contest web-page: Mark Davidson, phone: (225) 578-1581 contest@math.lsu.edu www.math.lsu.edu/~contest