Kósa András

English

University of Szeged

Acta Sei. Math., 45 (1983), 273—277 On the homotopy type of some spaces occurring in the calculus of variations A. KÓSA Dedicated to Professor B. Sz.-Nagy on the occasion of his 70th birthday 1. Let H€N and let D c R x R " be an open region. Suppose ¿̂ CR" are given such that (0, £0), (1, Denote by M(D) the class of continuous functions JC: [0, l]-«-R" such that (1) x(0) = £„, x(l) = fx, and r(x):= {(í, *(0)|*€[0,1])} c D. The space of Revalued continuous functions over [0,1] will be denoted by C„[0,1]. Thus M(D) is a subspace of C„[0,1]. Endow M(D) with the relative topology of C„[0,1]. The global methods of the calculus of variations (see [1], [3], [5] and [6]) lead us to the following problem: how can the homotopy type of M(D) be described from that of D1 In this paper we establish a connection between the homotopy types of the spaces D and M(D) for a rather wide class of regions D. We shall define a class of admissible regions and for this class we shall prove the following theorem. Theorem. Suppose D c R x R " is an admissible region and its homotopy type is the one point union S*i\/Sr*\¡ ...\¡S*k of the spheres Sr' of dimension (/ = 1,2, . . . , k). Then the homotopy type of M(D) is the one point union S"-i-iV5 r«-1V-VST"-1 of the spheres 5 r ' - 1 (i = l, 2,. . . , k). 2. In this section the necessary definitions and constructions will be given. Def in i t ion 1. The regions Dlt fi2cRn+1 satisfying (1) will be called r-in-variantly homeomorphic, if there exists a uniformly continuous homeomorphism <p:Di^-D2 such that a) <K(Uo) = «Uo)> = Receidev October 19, 1982. 18 A. Kosa b) the diagram <p: Di — D2 R is commutative where prx: R'XR"—R1 is the projection of the space R l X R " onto the first factor. n Denote by / „ cR" the n-dimensional open unit interval XK*> '[• Let k,i ¡=i (i=k) and r (r^n) be positive integers and <5£]0, l/2[ a real number. For the ordered quadruple (k, /', r, ¿) define the set Q(k, i, r, c5) as the product ( k m i x . i : Now, suppose that the positive integers n,k are given. Let r2, r(l)€N t for i = 1, 2, ..., k), a, p, 5£Ik. Suppose that for all / = 1,2, and 2d£Jk. The set D(k, r, a, P, ¿ ) c R X R " will be given in the following manner: D(k, r, a, p, 8) := {(/, *KRxR n | / € [0 , 1], x£i„, . and. if *£[<*„ p,] then x$Q(k, i, rt, <5,)}. D e f i n i t i o n 2. A region Z>cR"+ 1 is said to be admissible if there exist *€N,.r€N* (r^n, / = 1,2, ..., k), a, p, 5ak.(*i^P„ / '=1 ,2 , ..., k, 28<iln),.. such that k the intersection T l K ' f t t ¡ s nonempty, and D and D(k, r, a, p, 8) are' /-in-¡=1 • variantly homeomorphic regions. . . R e m a r k : It can be easily seen that the homotopy type of the regions D(k, r, a, p, 8) (and,thus that of D) is the one point union S'y\]S'*\/...\JST*. Now, choose real numbers <x0, p0,a.', P', t0 such that 0 < a ' < a 0 < / 0 < / ? 0 < / r < T . Define the function / : [0, 1] —[0, 1] in the following way: /€[0, a']U[P', 1],. te[p0, Pi-rn :={ t - a ! a „ - a - C o - a O , to, >o+4^r(p'~t0i P -P0 The homotopy type of some spaces 275 The restriction of / to the set [0, a0]U]/?o> U is invertible and the inverse also can be easily given: t, i Ç ^ a l U D S M ] , . ( / l [0 ,a 0 ]U ] / î „ , l ] ) 1 ( 0 = /—a a ' + r («0-oO> , tÇ[oc', /„], Q a P - to Let N, /-iN" ( r ^ n , / = 1 , 2 , f c ) , a', J3'€]0,1[ ( a ' c f t ) , <5eRk (2<5<E/*) be given. Define the subspace M(k, r, a', /T, <5)cC„[0,1] as follows: M(/c, r, a', P', §) .:= {x£ C„[0, 1] | * | [ s , . n const., | (x) = : x(a'), . Z(x)(LIn\nQ(k, i, r, d), x(t) = £O+4-(£(*)-£O) ('€[0, a']), ¡=i a *(O = ax)+-^r(Ql-ax)) ID}. . Finally, denote by j the identity map of [0, 1]. 3. We start with a simple observation. L e m m a 1. If the regions D1,D2czR"+1 satisfying the condition (1) are ^in-variant ly homeomorphic, then M(DX) and M(D2) are homeomorphic. P r o o f . Let <p\Dx-+D2 be a /-invariant homeomorphism (in this case, ob-viously, the inverse <p-1: D2—DX is also a /-invariant homeomorphism). Define the desired homeomorphism <P: M(D1)-~M(D2) as follows: (*(*)) (0 :=,pr2<p (/,*(/)) (/£[0,1]), . where pr2: R ^ R " — R " is the projection of the product space R1XR" onto the second factor. From the /-invariance of the homeomorphism <p it follows im-mediately, that <P is a homeomorphism. It is also clear that <P~1 has a form similar to that of ($-1(x))(/) = pr2<? )-1(i,x(0) (/€[0,1]). From Lemma 1 it follows that it is sufficient to determine the homotopy type .of; the spaces M(D{k, r, a, /?, <5)). We now turn to the calculation of the homotopy type of the space M(k, r, a.', ft', 5). For this purpose we shall prove the following L e m m a 2. The homotopy type of the space M(k, r;a.', P', 5) is the one point union 5 ' ' - 1 V5 r «- 1 V. . .V5 r " - 1 of the spheres S''"1 (/ = 1,2, ...,k). J . > • • "' . P roo f . It is obvious that the space M(k, r, a ' , P', <5) is homeomorphic to the n n-dimensional region 7 „ \ (J Q(k, i, r, 3). The desired homeomorphism W can be ¡=i 18« 276 A. Kosa given by (£ |Af(t,r,a',p',i))-\ where ^ is the function from the end of the 2nd k point. Now, by the definition of the sets Q(k, i, r, 5) the region / „ \ U U r-> >=i is homotopically equivalent to the one point union S T i - 1 V iS T i " 1 V . . .VS"* - 1 of the spheres S*"'-1 (i = 1 ,2 , . . . , Ac). * Choose numbers t0£ f ) ]a(, ft[ and a0 , ft; a', ft£]0, 1[ such that the inequalities <=i 0 < a' < a0 < mm {«¡} < max {ft} < ft, < ft < 1 are satisfied. L e m m a 3. The space M(k, r, a', ft, 5) is a deformation retract of the space M(D(k, r, a, ft <5)). P r o o f . A homotopy F: [0,1 ]XM(D(k, r, a, ft <5)) - M(D(k, r, a, ft 5)) is defined by _ f x o ( 2 t / + ( l —2x)j), t€[0, 1/2], F(x, x) : - | ( 2 _ 2 t ) ; c o / + (2T—1) !F(x(r0)), T€[1/2, 1], where r]/ is the function from the preceding proof. The restriction of X<-+F(T, X) to M(k, r, a', ft, <5) is the identity, because the elements of M(Jc, r, a.', ft', <5) are constant over [a', ft] and linear over the rest of [0,1], consequently xo(2T/+( l -2T)/ ) | [ a , , , , ] = £(*) = x \ [ a , < n , o(2t/ + (l —2T)/)|[o,a'juw,i] = * o (2r/+ (1 - 2T)j) j[0i x>]ulir,1} = X^.^uwm]» for x€[0,1/2], and [ (2—2x)x of+ ( 2 T — 1 ) !P (X ( I 0 ) ] [A ' , / ) ' ] = ( 2 - 2 T ) X ( / 0 ) + ( 2 T - 1 ) X ( / 0 ) = x | [ a , , r ] > [(2—2t) x o / + (2T—1) (x (/0)] [o, A- ] u w Ml = [(2-2T)X+(2T-1)X]|[0,C[.]l,[p..1i = for T6[l/2, 1]. The function xt—F(0, x) is the identity over M(D(k, r, a, ft <5)). The function XH-F(1,X) is a retract of M(D(k, r, a, ft <5)) onto M(k,r,u', p',5). The proof of Theorem follows immediately from Lemmas 1—3. If n=2 and the region D is 73 \{(i , 1/2, 1/2) | /€[0,1]}, furthermore eo =(1/2, 1/3), then the homotopy type of M(D) is the one point union V S® i= — » of infinitely many 0-dimensional spheres. There are as many spheres as there are different ways to wind the graphs of the functions around the omitted segment. The homotopy type of some spaces 277 References [1] M. MORSE, The calculus of variations in the large, Amer. Math. Soc. Colloquium Publ., vol. 18, Amer. Math. Soc. (New York, 1934). [2] SZE-TSEN Ни, Homotopy Theory, Pure and Applied Mathematics, vol. 8, Academic Press (New York—London, 1959). [3] J. MILNER, Morse Theory, Princeton University Press (Princeton, R. I., 1963). [4] R. S. PALAIS, Morse theory on Hilbert manifolds, Topology, 2 (1963), 299—340. [5] J . T . SCHWARTZ, Nonlinear functional analysis, Gordon and Breach (New York—London— Paris, 1969). [6] А. К о ш а , О многообразии управляемых процессов в задаче Лагранжа, Дифференциаль-ные уравнения, in print. DEPARTMENT OF ANALYSIS II EÖTVÖS LORÁND UNIVERSITY MÜZEUM KRT. 6—« 1088 BUDAPEST, HUNGARY 

On the homotopy type of some spaces occuring in the calculus of variations

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On the homotopy type of some spaces occuring in the calculus of variations

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