By Allan J. Sieradski
The remedy of the topic of this article isn't really encyclopedic, nor was once it designed to be appropriate as a reference handbook for specialists. really, it introduces the subjects slowly of their ancient demeanour, in order that scholars are usually not beaten by way of the final word achievements of a number of generations of mathematicians. cautious readers will see how topologists have steadily subtle and prolonged the paintings in their predecessors and the way so much strong rules achieve past what their originators estimated. To inspire the improvement of topological instinct, the textual content is abundantly illustrated. Examples, too a variety of to be thoroughly coated in semesters of lectures, make this article compatible for autonomous research and make allowance teachers the liberty to pick what they're going to emphasize. the 1st 8 chapters are compatible for a one-semester path regularly topology. the whole textual content is appropriate for a year-long undergraduate or graduate point curse, and gives a robust beginning for a next algebraic topology path dedicated to the better homotopy teams, homology, and cohomology.
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Additional resources for An Introduction to Topology & Homotopy
Our principal technique is to generalize the notion of a chart to a chart surface (or curtain) in dimension 3. In dimension 4, we develop a 3-dimensional analogue called an interwoven solid. We recall from  that a chart is a labeled ﬁnite graph in the plane that has three types of vertices: 1-valent black vertices, 4-valent crossings, and 6valent white vertices. The labels upon the edges incident at crossings and 6-valent vertices are required to satisfy additional conditions that we discuss below (Sec.
Xk+2 ) → (x21 − x22 , 2x1 x2 , x3 , . . , xr+2 ). A branched cover of degree n is also called an n-fold branched cover. Throughout, we will only deal with simple branched coverings, and so we will speak colloquially of branched coverings. We sometimes work in the PL-category. page 38 January 8, 2015 16:11 New Ideas in Low Dimensional Topology 9in x 6in b1970-ch02 How to Fold a Manifold 39 3. 2-Dimensional Simple Branched Coverings Let f : M 2 → S 2 be an n-fold simple branched cover with branch set L, and let f : M 2 \ f −1 (L) → S 2 \ L be the associated covering map; that is f is the restriction of f to the complement of the branch set.
The interwoven solid that yields its 3-fold branched cover is indicated as a sequence of curtains. Successive curtains in this case diﬀer from each other by replacing a chart move and its inverse with a product of charts or vice versa. So for each still in the movie (or braid movie) of the knotted surface, we construct a curtain. In the case of the 2-fold branched cover, the curtain is a Seifert surface. Each critical event for the knot movie induces a critical event between the curtains. Each will be described explicitly.