By Allan J. Sieradski

ISBN-10: 0534929605

ISBN-13: 9780534929602

The remedy of the topic of this article isn't encyclopedic, nor was once it designed to be appropriate as a reference guide for specialists. fairly, it introduces the subjects slowly of their old demeanour, in order that scholars usually are not crushed via the final word achievements of a number of generations of mathematicians. cautious readers will see how topologists have steadily sophisticated and prolonged the paintings in their predecessors and the way such a lot sturdy rules succeed in past what their originators predicted. To inspire the improvement of topological instinct, the textual content is abundantly illustrated. Examples, too a number of to be thoroughly coated in semesters of lectures, make this article compatible for self reliant examine and make allowance teachers the liberty to choose what they'll emphasize. the 1st 8 chapters are appropriate for a one-semester direction commonly topology. the whole textual content is acceptable for a year-long undergraduate or graduate point curse, and offers a powerful origin 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**

**Example text**

For the other vertices a/b , the example a/b = 5/8 is shown in the left half of the ﬁgure below. The ﬁrst edges drawn to this vertex come from 2/3 and 3/5 , and after this all the other edges from 5/8 are drawn in turn. As one can see, they are all obtained by adding (5, 8) to (2, 3) or (3, 5) repeatedly. If we choose any one of these edges from 5/8 , say the edge to 2/3 for example, then the edges from 5/8 have their other endpoints at the fractions (2 + 5k)/(3 + 8k) as k ranges over all integers, with positive values of k giving the edges on the upper side of the edge to 2/3 and negative values of k giving the edges on the lower side of the edge to 2/3 .

Recall the original example that we did: 67 − − − 24 19 = 2+ − − − − = 2+ 24 1 − −−− 24 / = 2+ 1 − − − − − −−− 1 + 5/ = 2+ 19 19 1 −− − − − −−1− 1 + − − − − − − − − 19 / 5 = 2+ 1 −− − − − −− − − − −− 1 1 + − − − − − − − − 3+ 4 /5 = 2+ 1 −− − − − −− − − − −− 1 1 + − −−−−−− − 1 3 + − − − − − 5/ 4 = 2+ 1 − − − − − − − − − − − − − − − − − − − − − − − − 1 1 + − − − − − − − − − − − − − − − − − 1 3 +− − − − − − − − − − 1 1 + − − − 4 The sequence of steps is the following: (1) Write x = a0 + r1 where a0 is an integer and 0 ≤ r1 < 1 (2) Write 1/r1 = a1 + r2 where a1 is an integer and 0 ≤ r2 < 1 (3) Write 1/r2 = a2 + r3 where a2 is an integer and 0 ≤ r3 < 1 and so on, repeatedly.

There is a close connection between the Diophantine equation ax+by = n and the congruence ax ≡ n mod b , where the symbol ≡ means “is congruent to". Namely, if one has a solution (x, y) to ax + by = n then ax ≡ n mod b , and conversely, if one has a number x such that ax ≡ n mod b then this means that ax − n is a multiple of b , say k times b , so ax − n = kb or equivalently ax − kb = n so one has a solution of ax + by = n with y = −k . Using this viewpoint, ﬁnd all integers x satisfying the congruence 31x ≡ 1 mod 71 , and then do the same for the congruence 31x ≡ 10 mod 71 .

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