From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Klein Bottle Date: 27 Nov 1996 14:27:39 GMT In article <57fk5v$6qj@gap.cco.caltech.edu>, Douglas J. Zare wrote: [in re: the image of the usual immersion of the Klein bottle into R^3] >Claim: The surface is the connected sum of two tori. There seems to be some disagreement about terminology here: the connected sum is an operation defined for two connected manifolds of the same dimension, and it results in another manifold. The space in question is not a manifold. (Connected sum is M1 # M2 =((M1 - B) union (M2-B))/S, that is, remove an open ball from each M, take the union and identify the corresponding points of the two boundaries, which are spheres. One needs to show that up to equivalence this is independent of the choice of the balls and the homeomorphisms between their boundaries, which is a little tricky since it isn't really true as stated.) One way to view the particular surface in question is as a quotient. Since the Klein bottle may be viewed as a quotient of the unit square: |---------->>--------------------- | | | |--->--| | ^ | v v | ^ | | | |----<-| | | | ----------->>---------------------| I've added an internal loop to the usual picture since we are identifying not only the opposite pairs of sides, but also identifying the left and right sides with the middle loop. This picture might make it easier to understand whatever interests you about the space -- cellular decomposition, fundamental group, Betti numbers, etc. dave