# An Introduction to Topology & Homotopy by Allan J. Sieradski

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Additional resources for An Introduction to Topology & Homotopy

Example text

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 [10] 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.