By Allan J. Sieradski
This article is an advent to topology and homotopy. issues are built-in right into a coherent entire and built slowly so scholars should not crushed. the 1st half the textual content treats the topology of whole metric areas, together with their hyperspaces of sequentially compact subspaces. the second one 1/2 the textual content develops the homotopy type. there are various examples and over 900 workouts, representing a variety of trouble. This publication can be of curiosity to undergraduates and researchers in arithmetic.
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Extra info for An introduction to topology and homotopy
10). 40 one easily checks that in this case the diﬀerentials xi · dx ωij = ; 2 ≤ j ≤ d − 1, 0 ≤ i ≤ j − 2 yj are holomorphic. To do that one ﬁrst notes that now the function x (resp. y) has d simple zeros at the points Pk = (0, ξdk ) (resp. Qk = (ξdk , 0)) and d simple poles at the points ∞k . Accordingly dx has zeros of order d − 1 at the points Qk and double poles at the points ∞k . 11. 1 Basic deﬁnitions 23 Clearly the diﬀerential dx (resp. the function y) has a zero of order 6 (resp. a simple zero) at the point P = (0, 0), another zero of order 6 (resp.
11). 12 we show how to repeatedly apply operation two followed by a single application of operation one to obtain a polygon with only one class of vertices. 13. We ﬁnd that the genus of the surface we started with is g = 3. Compact Riemann surfaces and algebraic curves 34 b a e c g d c e f f a b g d Fig. 11. The lines indicate the side pairings needed to build the topological surface corresponding to the word w = abcdef b−1 gd−1 a−1 f −1 c−1 g −1 e−1 . a b c d x y e e x g y g x cut along x y c paste along b d f a e c t a b g e g f f y a c d cut along t paste along g c y x y x e e e x x a y t cut along r r y t t r paste along a r t e Fig.
2 Topology of Riemann surfaces 29 −1 labels ai , bi or a−1 as described above. The above comment i , bi means that if we write down the sequence of labels we encounter when travelling, say, counterclockwise along the whole boundary ∂R, both symbols ai , a−1 (resp. bi , b−1 i i ) must occur in the resulting word w. 5), the simplest example of a non-orientable surface. Turn and glue e e e obius strip does not admit a Riemann surface structure. Fig. 5. 6). Then ϕ = Id serves as a chart around any interior point of the rectangle in both cases.
An introduction to topology and homotopy by Allan J. Sieradski