By Richard H. Crowell

Knot conception is one of those geometry, and one whose attraction is particularly direct as the gadgets studied are perceivable and tangible in daily actual house. it's a assembly floor of such various branches of arithmetic as crew concept, matrix thought, quantity thought, algebraic geometry, and differential geometry, to call a number of the extra renowned ones. It had its origins within the mathematical conception of electrical energy and in primitive atomic physics, and there are tricks this day of recent purposes in sure branches of chemistryJ The outlines of the fashionable topological thought have been labored out through Dehn, Alexander, Reidemeister, and Seifert nearly thirty years in the past. As a subfield of topology, knot conception varieties the center of quite a lot of difficulties facing the placement of 1 manifold imbedded inside one other. This ebook, that's an elaboration of a chain of lectures given via Fox at Haverford collage whereas a Philips customer there within the spring of 1956, is an try and make the topic available to all people. basically it's a textual content ebook for a path on the junior-senior point, yet we think that it may be used with revenue additionally by way of graduate scholars. as the algebra required isn't the accepted commutative algebra, a disproportionate volume of the e-book is given over to priceless algebraic preliminaries.

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And observe that if u '" v then 4>'(u) = 4>'(v). It follows that 4>' induces a homomorphism of F[d] into H. This homomorphism is clearly an extension of the function 4>: Cd] ->- H; thus Cd] is a free basis of F[d]. If now G is any group that is mapped onto F[d] by an isomorphism A, then E = A-l[d] is obviously a free basis of G, so that G must be a free group. Conversely let G be a free group, and let E be a free basis of G. Let F[d] be the free group on an alphabet d whose cardinality is the same as that of E.

It is the subgroup of G generated by all elements of the form gn. It should be clear that GO = 1, Gl = G, and that am c Gn whenever m is divisible by n. Also it may be noted that [G,G] c G2; in fact, glg2g1-1g2-1 = (glg2)2 . (g2- 1g1- 1g2)2 . g2-2, and this means that G/G2 is always abelian. If cp: G1 -+ G2 is a group homomorphism and W is any subset of a free group, then cp W(G 1) c W(G 2) since cpw E Q(G 2) for any wE Q(G1). Consequently, there is induced a unique homomorphism cp* such that ~ ~ G1/W(G1) ~ G2/W(G 2) is a consistent diagram.

G2 ~ G3 , then (1pcp)* = 1p*cp*. (c) If cp is onto, so is cp*. (d) If cp is an isomorphism onto, so is cp*. 48 Chap. IV PRESENTATION OF GROUPS For example, if (a) and (b) have been verified, (d) follows by simply observing identity = (cpcp-l)* = cp*(cp-l)* = (cp-Icp)* = (cp-l )*cp*. Notice that if (d) is altered by the omission of the word "onto," the result is false. 1), we can prove a result that was promised in Chapter III. 2) If m and n are distinct cardinal numbers, then the free groups of rank m and n are not isomorphic.