G. Whitehead, "Elements of Homotopy Theory", Springer-Verlag, NY, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= h 1 h 2 . h n = idY so that all the latter are trivial on X. A d space is d contractible if it is d homotopy equivalent to a point, or equivalently if it admits a deformation retract at a point. Classical relations between ordinary homotopy equivalence and deformation retracts can be seen in [24], I.5. Plainly, all these relations imply the usual ones, for the underlying spaces. In fact, they are strictly stronger. As a trivial example, the d discrete structure c 0 R on the real line (where all d paths are constant, 1.1) is not d contractible. Less trivially, within path connected ....

G.W. Whitehead, Elements of homotopy theory, Springer, Berlin 1978.

....exact sequence of the triple for cohomology with local coecients (see for example [14] it follows by induction that H i Ln ; T ; gj Ln ) # j (Y ) 0 for all j 2 and all i 6 j. The generalization of Milnor s theorem on the cohomology of an expanding union to local coecients (see [19], Theorem 2.10 ) yields an exact sequence 0 lim 1 H i 1 (Ln ; T ; Gn ) H i (X; T ; G) lim H i (Ln ; T ; Gn ) 0; where Gn = gj Ln ) # j (Y ) and G = g # j (Y ) This shows that H i X;T ; g # j (Y ) 0 for all j 2 and all i 6 j. Hence the path ....

George W. Whitehead, Elements of homotopy theory, Springer-Verlag, Berlin-Heidelberg, 1978.

....(X; Z) 0 for 0 i n. For n Gamma1 the conditions of n connectedness and n acyclicness are vacuous. Theorem 2.1 (Hurewicz) For n 0, a topological space X is n connected if and only if the reduced homology groups H i (X; Z) are trivial for 0 i n and X is 1 connected if n 1. Proof. See [25], Chap. IV, Corollaries 7.7 and 7.8. Let X be a partially ordered set or briefly a poset. Consider the simplicial complex associated to X , that is the simplicial complex where vertices or 0 simplices are the elements of X and the k simplices are the (k 1) tuples (x 0 ; x k ) of ....

....Let X be a path connected space with a base point x and let F be a local system on X. Then the inclusion fxg , X induces an isomorphism F(x) G Gamma H 0 (X; F) where G is the subgroup of F(x) generated by all the elements of the form a Gamma fia with a 2 F(x) fi 2 1 (X; x) Proof. See [25], Chap. VI, Thm. 2.8 and Thm. 3.2. We need the following interesting and well known lemma about the covering spaces of the space jX j, where X is a poset (or more generally a simplicial set) For a definition of a covering space, useful for our purpose, and some more information, see [17, Chap. ....

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Whitehead, G. W. Elements of Homotopy Theory. Grad. Texts in Math. 61, SpringerVerlag, (1978). e-mail: mirzaii@math.uu.nl vdkallen@math.uu.nl

....the n th skeleton i.e. the immersion can be further extended to a second extension, and so on then the original cochain is said to be realizable or extendible. If one can reach the s th stage one says that the cochain is s extendible. We adapt to this context Eilenberg s obstruction theory, see [19], xV.5. A cochain in C k (M;P k ) can be thought of as a map k de ned from M k to QRP 1 , that restricted to any k cell is geometrically represented by an element of P k . The problem to which we apply Eilenberg s theory is that of extending this map over the next skeleton. 5.1 A review of ....

....Given a s simple space Y , a CW complex X and a map : X s Y the obstruction to extending f to the (s 1) skeleton is a cochain c s 1 ( 2 C s 1 (X; s (Y ) assigning to each (s 1) cell e s 1 the map j e s 1 . Its fundamental properties are stated in the following theorem (see [19], xV.5) Theorem 5.1. 1. is extendible to X s 1 if and only if c s 1 ( is the trivial cochain. 2. c s 1 ( is a cocycle, hence represents an element of H s 1 (X; s (Y ) 3. j X s 1 is extendible to X s 1 if and only if c s 1 ( is trivial in H s 1 (X; s (Y ) The problem ....

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G. W. Whitehead, Elements of Homotopy Theory. Graduate Text in Math., Springer, 1978. 23

....Theorem 2.7 (Robinson, Theorems 4.3.4 and 4.3.5 in [2] If G is a locally soluble group then r 0 (G) r(G) Moreover r(G) is bounded by a function of r 0 (A) and r 0 (B) Let us now derive some consequences, as general as possible, of the above results. It is well known (cf. Theorem X, 3. 6 in [17]) that [X; aut 1 (Y ) is a nilpotent group of class c when the Lusternik Schnirelmann category of X is c. In particular, when X is a coH space [X; aut 1 (Y ) is abelian. To avoid repetition, the standing hypotheses in all theorems is that the self equivalences of X Theta Y can be ....

.... Gamma] are as above. 19 Note that when m n Coker( n ; Gamma] m 1 (S n ) m n (S n ) m n (S n ) m 1 n 2n (S n 1 ) m 1 = n Indeed, if m 1 n, then m 1 (S n ) 0 and the claim follows from the Koh s exact sequence. When m 1 = n, then Corollary XII, 2. 6 of [17] implies that the image of [ n ; n (S n ) 2n Gamma1 (S n ) equals the kernel of the sospension homomorphism E : 2n Gamma1 (S n ) 2n (S n 1 ) which is onto by the Freudenthal s theorem. Therefore Coker[ n ; ImE = 2n (S n 1 ) 2. Products of Moore spaces. The ....

Whitehead G.W.: Elements of Homotopy Theory, GTM 61, Springer, 1978 24

....some definitions and notation: Definition 2.1. A Pi algebra is a graded group G = fG k g 1 k=1 (abelian in degrees 1) together with an action on G of the primary homotopy operations (i.e. compositions and Whitehead products, including the 1 action of G 1 on the higher G n s, as in [W, X, x7]) satisfying the usual universal identities. See [Bl3, x3] or [Bl1, x2.1] for a more explicit description. A morphism of Pi algebras is a homomorphism of graded groups OE : G G 0 which commutes with all the operations. Pi algebras form a category, which will be denoted Pi Alg. ....

G.W. Whitehead, Elements of Homotopy Theory, Springer-Verlag Grad. Texts Math. 61, Berlin-New York, 1971.

....equivalent to a simply connected CW complex, and a nonzero class w 2 H 2 (BH;Z 2 ) is specified (corresponding to a choice of double cover of H ) We can recover the classical theory by setting BH = BSO(n) n 2) with w the unique nonzero class w 2 H 2 (BSO(n) Z 2 ) Z 2 . Recall [8] that any map f : X Y can be transformed into a fibration by replacing X by the space P of paths from X to Y in the mapping cylinder of f . The initial point fibration p 0 : P X has contractible fiber, and the endpoint fibration p 1 : P Y is homotopic to f ffi p 0 . The fiber F of p 1 is ....

G. Whitehead, Elements of Homotopy Theory, Grad. Texts in Math. 61, Springer-Verlag, Berlin (1978) 10

....that many standard proofs in homotopy theory extend directly (via induction over n) to the category of n ads. In particular, the following hold by repeated use of the homotopy extension property for cofibrations, and standard techniques in manipulating mapping spaces (for proofs see, for example, [40]) 1. Maps: X Theta Y ) Z: Maps: Y: Maps: X: Z: adjunction) 2. f: 2 X: Y: is a cofibration if each fA is a cofibration of spaces, 3. If f: 2 X: Y: is a cofibration then f: Z: X: Z: Y: is a fibration. 4. Composition of maps provides a monoid structure for aut X: and a ....

.... of f [ 1 2 ] C 1 = M [ 1 2 ] and C 2 = N [ 1 2 ] Since k[ 1 2 ] and [ 1 2 ] are homotopy equivalences such that f [ 1 2 ] ffi k[ 1 2 ] 1 2 ] ffi f [ 1 2 ] there exists a homotopy equivalence t: 2 B aut (C: with t 1 = k[ 1 2 ] and t 2 = 1 2 ] see, for example, [40]) By construction, since ( k[ 1 2 ] 2 Gamma1) is nilpotent on H (M [ 1 2 ] and ( 1 2 ] 2 Gamma1) is nilpotent on H (N [ 1 2 ] t ; 2 Gamma 1) will be nilpotent on H (C ; We can therefore apply Theorem 2.4 to C: and t: 2 B aut (C: This produces a fibration C: ....

G. W. Whitehead, Elements of homotopy theory, Grad. Texts in Math., vol. 61, Springer-Verlag, Berlin, 1978.

....this paper, we will need to understand various Bockstein homomorphisms on the cohomology of BZ=p. A necessary first step is to know the cohomology of BZ=p with various coefficients. The integral homology or cohomology of BZ=p can be found either by computing it or by referring to sources such as [Wh]. In either case, we find that Hn(BZ=p) 8 : Z n = 0 Z=p n 0 and odd 0 otherwise (2.2) CHAPTER 2. BACKGROUND 9 and H n (BZ=p) 8 : Z n = 0 Z=p n 0 and even 0 otherwise. 2.3) An application of the universal coefficient theorem gives us: Hn(BZ=p; Z=p i ) 8 : Z=p ....

....spectral sequences, this ones indetermediate steps tell us something useful: A cycle which survives to E r n and is contained in the image of d r comes from a direct summand Z=p r of H n (X ; Z) For more on exact couples and CHAPTER 2. BACKGROUND 10 spectral sequences, see [McC] MT] or [Wh]; for more on the BSS, see [B] or [MT] There are, however, technical difficulties in using this form of the BSS for our purposes, which we will not elaborate on. In [MT] an alternate procedure is given. While for simplicity [MT] only discusses it for p = 2, a careful check shows the analogous ....

Whitehead, G., Elements of Homotopy Theory, Springer-Verlag, 1978.

....structure. Then equation (6) becomes 0 = x y) Delta Gamma [y; x] Delta H 2 ( Delta ae = x y) y; x] Delta (x Delta Gamma y Delta H 2 ( Delta ae: 7) Let f = x Delta ; g = Gammay Delta H 2 ( SigmaA Gamma SigmaW SigmaW . Using the Hilton Hopf expansion [Wh], which follows by naturality from equation (3) we have (f g) Delta ae = g Delta ae [g; f ] Delta H 2 (ae) g; f ] f ] Delta H 3 (ae) Delta Delta Delta : 8) By metastability, H 2 ( desuspends to an element i 2 [A; W [2] so by naturality [g; f ] Gamma[y; x Delta ] ....

....implies that ae and ae 0 are homotopic. We have now completed the proof of the existence part of Williams s Theorem 0.1, modulo the proof of Theorem 1.6. 2. A Review of Elementary Unstable Homotopy Theory We would like to rewrite the literature we survey here from the point of view of Whitehead [Wh], James [Ja1, Prop. 6.42] and Moore and Neisendorfer [M N] of working in the category of compactly generated spaces and NDR pairs, and using the result of Str m [St] Wh, thm. I.7.14] that the pullback under a fibration of an NDR pair is an NDR pair in the total space. We contend that only with ....

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Whitehead, G. W.: Elements of Homotopy Theory. (GTM). Springer-Verlag 1980

....gb . For a map f : A Gamma Omega 2 X, we have the adjoints fb : SigmaA Gamma Omega X and fb b : Sigma 2 A Gamma X. We now recall from Boardman and Steer [B S] the definition of smash products, cup products, James Hopf invariants and Whitehead products. See also Whitehead s book [Wh] for its attention to point set topology, and for the James splitting, which is not treated in [B S] or in any of James s papers. Suspension will mean smashing on the right with S 1 = I=f0; 1g, so SigmaX = X S 1 . We define S n = S 1 ) n] I n = I n ) By associativity of the ....

Whitehead, G. W.: Elements of Homotopy Theory. (GTM). Springer

....a well developed theory of affine bundles and we can only usefully categorise it as a fibration. A fibration is a map p : E M of topological spaces that is onto and has the Homotopy Lifting Property for all topological spaces Y that are mapped to the base through a map f : Y f M see [13], p.29. In [11] we present another fibration associated to that of the control indicatrix that is of crucial importance to the solution of the general problem of control. Now it is useful to have available explicit negative results concerning nonlinear controllability accessibility. This is ....

George W. Whitehead, Elements of Homotopy Theory, Springer, GTM 61, 1978.

...., provided the suspension is unreduced. With these conventions fact 2.5 and corollary 2.6 hold for n = 1. 3. Proof of the main theorems Let X 1 ; X k be sufficiently nice topological spaces, for example connected CW complexes with non degenerate basepoints. By the Hilton Milnor theorem [10] there is a natural homotopy equivalence h : Y i2I Omega SY i Omega S k j=1 X j ; 2) where I denotes the set of basic products on k letters. Essentially, each Y i is a smash product of some of the spaces X 1 ; X k corresponding to the basic product i. By the formula due to ....

G.W. Whitehead, Elements of homotopy theory, Springer-Verlag, New York, (1978).

....two A modules V and W a homomorhpism : V W is a collection of linear operators x : V x W x , x 2 P such that for any path a with the initial point x and terminal point y holds y a(V ) a(W ) x . 3 Definition of the algebras Let K be a regular cell decomposition of the sphere S n ([15]) n 0. We will denote by K i , 0 i n the set of i cells in K. Consider a poset P = P (K) with elements ; L, e 2 K i , 0 i n and the relation defined as follows: 1. x for any ; 6= x 2 P ; 2. x L for any S 6= x 2 P ; 3. x y, x; y 2 K if and only if x 2 y. For the rest of the ....

....(x) Gamma1 i n 1. By lemma 1 any homomorphism : C i C i 1 , Gamma1 i n is uniquely defined by a complex matrix d = d yx ) x2P i y2P i 1 : Extend our cell complex K to cell decomposed ball K 0 by adding one n 1 cell L in a natural way. Fix an orientation for each x 2 K 0 ([15]) and set [x : y] be the incidence numbers with respect to this orientation. Denote by d i = d yx ) x2P i y2P i 1 ; 0 i n the complex matrix such that d yx = y : x] We also set d Gamma1 = d x; x2P0 ; d x; 1 for all x 2 P 0 : As it was mentioned d i , Gamma1 i n defines some ....

G.Whitehead, Elements of Homotopy Theory, Springer-Verlag, Ney York, 1978.

....deformation we may achieve the same effect in 3 d , addressed below, but for now our intuition with Euclidean space will suffice. In topological language, any two paths with common endpoints in such a simply connected space are automatically homotopic, an equivalence relation on paths [Bro88, Whi49, Whi78]. They become nonhomotopic when a hole appears somewhere in the space between them to inhibit their deformation into one another. Homotopy is ordinarily studied for spaces the movements in which form a group under composition, where homotopy is inhibited only by holes. The typical irreversibility ....

G.W Whitehead. Elements of Homotopy Theory. Springer-Verlag, 1978.

....local coe#cient system Z#(X) such that Z#(X) x) #X Z#(X) x # ,x) for any x X.Let# n (X) be the covariant (or contravariant) local coe#cient system on X determined by the nth homotopy group functor for n 2. Then we get the following statements the proofs of which follow from e.g. [20, 24]. Y of CW spaces the following assertions are equivalent: 1) the map f is a weak equivalence; 2) the map f : K,X] K, Y ] of homotopy classes is a bijection for any CW spaces K; 3) the maps P n f : P n X P n Y are weak equivalences for all n #(Y ) is an equivalence of ....

G.W. Whitehead, Elements of homotopy theory, Springer-Verlag,

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G. Whitehead, "Elements of Homotopy Theory", Springer-Verlag, NY, 1978.

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G. Whitehead, "Elements of Homotopy Theory", Springer-Verlag, NY, 1978.

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G.W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Math., Springer,

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G.W. Whitehead, Elements of homotopy theory, Grad. Texts in Math. 61, Springer, 1978.

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G.W. Whitehead, Elements of homotopy theory, Grad. Texts in Math. 61, Springer, 1978.

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G. W. Whitehead. Elements of Homotopy Theory. (Springer-Verlag, 1978).

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Whitehead, G.W.: "Elements of homotopy theory", Graduate Texts in Mathematics, Springer (1978).

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G. W. Whitehead, "Elements of Homotopy Theory," Springer-Verlag, New York, 1978.

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G. Whitehead. Elements of Homotopy Theory. Springer-Verlag, New York,

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