note on a previous paper entitled on adding relations to homotopy groups

Annals of Mathematics

Note on a Previous Paper Entitled On Adding Relations to Homotopy GroupsAuthor(s): J. H. C. WhiteheadSource: Annals of Mathematics, Second Series, Vol. 47, No. 4 (Oct., 1946), pp. 806-810Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1969237 .

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ANNALS OF MATHEMATICS

Vol. 47, No. 4, October, 1946

NOTE ON A PREVIOUS PAPER ENTITLED "ON ADDING RELATIONS TO HOMOTOPY GROUPS"

By J. H. C. WHITEHEAD

(Received November 20, 1945)

1. The group hr The object of this note is to fill a gap in the definition of, and to make certain

remarks concerning a group, hr, which was introduced in H. G. (p. 421).1 The definition should read as follows:

Let r be a given group, whose elements we denote by small greek letters, and let H be an aggregate of individuals a, a-', b, b-1, *, such that a # a-', b 5 bV' etc.2 Let h(H) C F be a single-valued, but not necessarily (1 - 1), transformation of H in F, such that h(a-') = {h(a) }-'. We denote by hr the group generated by all pairs (a, i), to be denoted by at, with (a, 7 ' = (a-', ), subject to the relations

(1.1) atbt,= brat, for each a, b C H, i, il C F, where r = th(a)t7 'n (if a = b, t = l, this reduces to at = ath(a)).

We recall from H. G. the definitions of the automorphism Oi(hr) = hr. defined by

O,(at) = are,

for any r E F, the homomorphism3 h(hr) C F, defined by

h(at) = th(a)t-',

and the general form of the relations (1 - 1), namely

(1.2) xyx = Oh(z) (Y),

which is a formal consequence of (1 - 1) for any x, y C hr. Since rh(x)r'1 = h{OT(x) }, for any x E hr, r E F, the sub-group h(hr) is in-

variant in F.

2. The relative homotopy group 7r2(X*, X) As in H. G. let X be an arcwise connected topological space and let

X* = X + E 2 + E 2 + ,

where El, E2, * are oriented 2-cells, bounded by oriented circuits in X and not meeting X or each other in any point which is internal to any of them.4

1 Annals of Math., 42 (1941), 409-28. This paper will be referred to as H. G. 2 We might have described H as the aggregate of free generators, with their inverses,

of a free group, and the following transformation, h, as defining a homomorphism of this free group in r. We did not do this because the corresponding group multiplication in the free group plays no part in what follows.

3 In H. G. h(h ) was denoted by 4(hr). 4Here we remove the unnecessary assumption that these cells are finite in number,

which was implicit in H. G.

806



ON ADDING RELATIONS TO HOMOTOPY GROUPS 807

Let ti C X be a segment joining a base point x0 E X to a point xi on the boundary of E2, let r, be the element in the relative homotopy group5 7r2(X*, X), which is represented by the singular cell ti + E~ , and let a, E 7r,(X) be the element corresponding to the boundary of this cell. Let c E X be any oriented circuit, which begins and ends at xo and let t be the corresponding element of 7r1(X). Then the element of 7r2(X*, X), which corresponds to the singular cell c + ti + Es, is the image of ri in an automorphism, At, of 7r2(X*, X). If the map f(S2 po) C X*, of H. G., p. 422, is replaced by a map f(E2, Po) C X*, such that f(E2, po) C X, f(po) = xo, where E2 is a 2-element and Po e A2 then the argu- ments leading up to (6.5) in H. G. show that any element in 7r2(X*, X) is expressible as a product of the elements Ae(ri) (i = 1, 2, * ).

The proof of Theorem 4 in H. G., and the argument in H. G., pp. 425, 426, with only trivial modifications, serve to establish the theorem:

Theorem. If H = (ar , alr, ***), with one symbol a1 corresponding to each cell E~, and h(ai) = ei, then the transformation

#Ae(ri) = aqe

determines an isomorphism of 7r2(X*, X) on hr which is (obviously) an operator isomorphism, in the sense that

= Orat',

for any Ere ,r(X). If X is the natural homomorphism of 7r2(X*, X) in 7r,(X) it is clear that

X = hip.

In terms of this theorem the isomorphism between h'1(1) C hr and the residue group 7r2(X*, xo) - 72?, with which Theorem 4 of H. G. is concerned, follows from a general theorem6 concerning the groups 7rr(X*, X), 7rr(X*, Ao), 7rr(X, XO), 'Fr-1 (X, x0).

As in the case of absolute homotopy groups' it follows that, if 3-cells E3, E3, ... are added to X*, to form a space X3, in the same way that E 2, E2, ...

were added to X to form X*, then the relative homotopy group 7r2(X3, X) is obtained from 7r2(X*, X) by the addition of relations u1 = 1, u2 = 1, * , where u1 e -1(1) is the element corresponding to the boundary of the cell E . Thus 7r2(X3, X) may be represented as the group generated by at, be, * -, subject to the relations (1.1) and xi = 1 (i = 1, 2, ... ), where xl, x2, * are certain elements in h-1(1).

3. The group h-1(1) when X = K1 The purpose of this section is to show that, when X* is a 2-dimensional com-

plex, K2, and X = K1, its 1-dimensional skeleton, then the study of the group 5 See W. Hurewicz and N. E. Steenrod, Proc. Nat. Academy of Sciences, 27 (1941),

60-64; also J. H. C. Whitehead, Proc. L. M. S., 48 (1944), pp. 281 et seq. 6 J. H. C. Whitehead, Proc. L. M. S., loc. cit., Theorem 13, p. 285. H. G., Theorem 3

can also be stated conveniently in terms of relative homotopy groups and this general theorem.

7 J. H. C. Whitehead, Proc. L. M. S., 45 (1939), Theorem 18, p. 281.



808 J. H. C. WHITEHEAD

h'(1) leads straight back to the known isomorphism between 7r2(K2) and the homology group 132(R2), where 1?2 is the universal covering complex of K2. Thus there is less novelty of algebraic method than I originally thought there was in the definition of 7r2(K2) by means of the group hr.

Let F be the free group wr(K1). Then h(hr), being a sub-group of F, is itself a free group.8 Let {l, 6 , * * - be free generators of h(hr) and let xl, x2, ** C. hr be elements such that h(xi) = {;. Then a relation

Xtl *-X~t=1 f -gl E S

implies

whence { '. ** reduces to the empty product by cancelling out consecutive terms of the form t ! . Therefore the same applies to x* *** x* and it follows that xi, X, * ... are free generators of a free sub-group Fo C hr, and h induces an isomorphism hO(Fo) = h(hr). Let y be any element of hr and let

h (y) = ... = W ) .

Let Wv(x) = h-oh(y). Then

y = W.(x) WY (x)y and

h{W1 l(x)y-.= {h(y)l}'h(y) = 1.

Thus any element y e hr is expressible in the form

y = xo 8,

where xo e Fo and h(s) = 1. This expression is unique, since xo s = 1 implies 1 = h(xo)h(s) = h(xo), whence xo = 1 and hence s = 1. It follows from (1.2) that s is in the centre of hr , whence hr is the direct product

(3.1) hr = Fo X hT1(1).

Let Ab(G) denote the "abelschgemachte" group G/C(G), where G is an ar- bitrary group, and C(G) is its commutator group. It follows from (3.1) and the fact that h-'(1) is abelian, that

Ab(hr) = Ab(Fo) X h7 '(1).

Also it is clear that the transformation

ha(i) ={i

determines an isomorphism of Ab(Fo) on Ab{h(hr)1, where xi and tj are the elements of Ab(Fo) and Ab h(hr) which correspond to xX and {i. If we extend h. to a homomorphism ha. Ab(hr) } = Abth(hr) } by writing

ha(2s0) = ha(Xi) (s e

8 Q. Schreier, Abh. Math. Seminar Hamburg, 5 (1927), 161-183.



ON ADDING RELATIONS TO HOMOTOPY GROUPS 809

we have h'(1) = ha'(1).

Therefore, in attempting to calculate 7r2l(K), we may make hr and h(hr) abelian to start with. - If we add the relations xy = yx to (1.2) it follows that

(3.2) y -oh() (=),

for any x, y C hr. Conversely, (3.2) and xy = yx imply (1.2). Therefore we may represent Ab(hr) as a free abelian group, freely generated by at, b-,, **, where I, q C r/h(hr) = 7r,(K2). Moreover, if we rewrite Ab(hr) with addition instead of multiplication, writing air =a , we have

el e

ale * ** are. = El a, + * + E?~T ar.

Therefore, if we associate air with the 2-cell tM C K2, which covers E2 and is determined by base points xo e K2, lo e K and a singular segment,9 c + ti C K', joining xo to x e F(E), we may identify Ab(hr) with the group of 2-chains in K2, with integral coefficients.

Let Ko be the 1-dimensional skeleton of j2. Then Rol is the covering of K', which is determined by the (invariant) sub-group h(hr). Therefore h(hr) may be identified with 7r,(Kl), which is then freely generated by t1, 2,*

The boundary, in the sense of homotopy, of the cell 2t c K2, when joined to .o by the segment covering c + t,, represents the element h(aq). When multiplication is made commutative, 41, 6, *** bcome free generators of the homology group lB(Kf), and ha(ai) represents the boundary, in the sense of homology, &(th2). Therefore ha(x) is the (homology) boundary of x, for any x e Ab(hr), and h'a1(l), or haj(0) if addition is used, is the group of 2-cycles in R2. Hence we are back at the familiar approach to 7r2(K2) by means of the isomorphism

72(K2) - I2(K2)

and Reidemeister's theory of tberlagerung.'0

4. Errata to H. G. I take this opportunity of correcting the following misprints in H. G.

p. 414, 1. 4-For S' read S2. p. 415, 1. 3-For y1(t) = y read 4,(y) y. p. 419. 1. 17 up-For "f-1(A') is a * read "f-'(A ) is a p. 422, 1. 7 up-For ".-- and we shall have x = 4p(f, 2)." read "--a of such

a kind that x = t{f, 3)." p. 423, 1. 7-For " in a linkage* *" read " is a linkage.

1. 12 up-For " - - circuit 4, + c, + 4, ." read " -* circuit 1, + c,-Ip .-

11. 4 up to bottom-For px, pa etc. read p"', p,, etc.

9 Cf. ?2 above. 10 For references see H. G.



810 J. H. C. WHITEHEAD

p. 425, 1. 3-For " *replace c by c .." read " * . . replace c by c, ...

1. 4-For g of air " read " g or ait ." 1. 9- For 0 =() 1q read = =0(x) last line-For aa-'bb-' read abb-'a-1.

p. 426, 1. 15 up-For A' = Al read A' = Al. p. 427. 11. 5-9-Among the relations between the elements of 4-1(1) should be

included the multiplication table for 4-1(1) itself. These can be enumer- ated constructively with the elements of 4-1(1), since each new element can be tested to discover if it is a product of any two which precede it, or their inverses (note that, in this passage, 0 is a homomorphism of the free group F, not of its factor group h11).

BALLIOL COLLEGE OXFORD



note on a previous paper entitled on adding relations to homotopy groups

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