By Stephen Huggett

This is a ebook of undemanding geometric topology, during which geometry, usually illustrated, courses calculation. The ebook begins with a wealth of examples, frequently sophisticated, of ways to be mathematically sure even if gadgets are an analogous from the viewpoint of topology.

After introducing surfaces, resembling the Klein bottle, the booklet explores the homes of polyhedra drawn on those surfaces. extra subtle instruments are built in a bankruptcy on winding quantity, and an appendix offers a glimpse of knot idea. in addition, during this revised variation, a brand new part provides a geometric description of a part of the category Theorem for surfaces. numerous impressive new photos exhibit how given a sphere with any variety of usual handles and at the least one Klein deal with, all of the traditional handles may be switched over into Klein handles.

Numerous examples and routines make this an invaluable textbook for a primary undergraduate path in topology, offering an organization geometrical starting place for extra research. for a lot of the e-book the necessities are mild, even though, so somebody with interest and tenacity can be capable of benefit from the *Aperitif*.

"…distinguished by way of transparent and beautiful exposition and weighted down with casual motivation, visible aids, cool (and fantastically rendered) pictures…This is a very good ebook and that i suggest it very highly."

MAA Online

"*Aperitif* evokes precisely the correct impact of this publication. The excessive ratio of illustrations to textual content makes it a brief learn and its enticing type and material whet the tastebuds for a number of attainable major courses."

Mathematical Gazette

"*A Topological Aperitif* offers a marvellous creation to the topic, with many various tastes of ideas."

Professor Sir Roger Penrose OM FRS, Mathematical Institute, Oxford, united kingdom

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**Additional resources for A topological aperitif**

**Example text**

S. Huggett, D. 1007/978-1-84800-913-4 3 c Springer-Verlag London Limited 2001, 2009 25 26 A Topological Aperitif This example helps us to see that what we are studying in this chapter is something more complicated than just a set. It is a set within a set. 1 are equivalent subsets in space, but non-equivalent subsets in the plane. 2, although this is actually a harder example: we postpone further discussion to Appendix B. 2 Our deﬁnition of the equivalence of subsets is given in terms of homeomorphism, so that, as with homeomorphism itself, we work with the idea of a correspondence rather than a deformation.

The resulting set in R4 looks the same as 54 A Topological Aperitif before, because no x, y or z coordinate has been changed, but is now really a Klein bottle: the thin tube does not intersect the thick tube. The Klein bottle is non-orientable: if we start on the “outside” and go once round the Klein bottle we end up on the “inside”. Unlike the sphere, then, which has a well-deﬁned inside, the Klein bottle has no inside. Now cut the Klein bottle into two congruent pieces by the plane passing through the centre of each circular section of the tube.

The plane set S is path-connected and is the union of three line segments, each segment being not only homeomorphic to [0, 1] but also straight. Find eighteen examples of such a set S, no two of your examples being homeomorphic. Show that no two of your examples are homeomorphic. 3. The plane set S is path-connected and is the union of the axes and a circle (a round circle, not just a set homeomorphic to a circle). Find eight such sets S, no two being homeomorphic. Show that no two of the sets are homeomorphic.