On the Maunder type theorem in the ex-homotopy category

We study the homotopy theory of comma category and define the cell structure and Postnikov system in the ex-homotopy category. By using these structures we give the four types of spectral sequences and show that Maunder type theorems hold for these spectral sequences. Introduction In [5], C. R. F. Maunder defined the cohomology spectral sequence associated with the Postnikov decomposition of £? -spectrum of target object and showed that his spectral sequence coincides with Atiyah-Hirzebruch spectral sequence. In [4], T. Matumoto proved Maunder's theorem in the equivariant homotopy category. We remarked in [7] that Maunder's theorem holds also in the category of functor complexes. In [6], we studied the unstable version of Maunder's theorem and applied them to the theory of phantom maps. Thus it is interesting to know whether Maunder type theorem holds in a homotopy category. In this paper, we define homotopy spectral sequences associated with cell structure and Postnikov system and prove Maunder type theorems in the ex-homotopy category. In § 1, we study the homotopy theory of comma category and obtain results analogous to the ones of the ordinary homotopy theory (e. g. J. H. C. Whitehead's theorem). In §2, we define the cell structure and Postnikov system in the ex-homotopy category and obtain the duality between them. In §3, we define the homotopy spectral sequences associated with the cell decomposition of a source object and the Postnikov decomposition of a target object by the same way as [6]. In this paper, we shall show that these homotopy spectral sequences are isomorphic as exact couples. Moreover analogously we define the homotopy spectral sequences associated with the anti-skeleton filtration and anti-Postnikov decomposition defined in § 3. We also prove that these are isomorphic as exact couples. Communicated by N. Shimada, September 14, 1988. Revised December 19, 1988. * Kurume Institute of Technology, Kurume 830, Japan. 632 YOSHIMI SHITANDA § 1. The Homotopy Theory of Comma Category We review the abstract homotopy theory defined in [7], In this paper, we shall use the results in [7] and terminologies and notations in S. MacLane [3]. Definition 1.1. We call a category C a pre-homotopy category if it satisfies the following axioms (Al-3). (Al) C is closed under finite limits and finite colimits; hence it has the initial object 0 and the terminal object 1. (A2) There are given covariant functors /, P: C-+C with a natural isomorphism C(IA, B}=C(A, PB} for any objects A, B of C (C(—, —) is horn-set in C). We call these the cylinder and path functors respectively. (A3) Moreover there are three natural transformations k*°. Id—*I (k=Q, 1) and T: I-^Id with rO#=/rf=rl*. Here Id means the identity functor or identity natural transformation. 0#, 1* and r are called the top-face, bottom-face and projection transformations respectively. Let I be n-time composed functor of / ( P — I d } ; and define the natural transformations dj=I-'k*I': /->/ and s,=/-r/>: I-+I for (/, fe)e= [n]X[l] ([m]={0, 1, ••• , ra}). We call these the face and degeneracy operators respectively. These operators d j k , sj satisfy the cubical simplicial relations (cf. Lemma 1.3 in [7]). Let (i'0, fe0)e[n]x[l]. Then by patching the 2n+lfaces dt*: I->I for (f, k)=£(i0, k0) according to the cubical simplicial relations, we have the functors J—J(iQ, kQ) and the natural transformation ^:/ -»/0 We use the letter J for any (/0, *0) (J°=Id). Now we consider the following extension condition and the natural homotopy axioms for a pre-homotopy category C: (E. C) For any morphism f:JX—>Y, there is a morphism F: IX-*Y with FJi=f. (NHA 1) There is a natural transformation f j t : I n * J n 1 with pji=ld for all n>0, that is, (EC) holds naturally by taking Fp=f. (NHA 2) There is a natural transformation p: l-»j~ with ri^=rn and fjL%=Id for all ? 2>0, where rn\ I -*Id and r'n: J ~^Id are defined by compositions of projections r. Let C be a pre-homotopy category. We call C an abstract homotopy category if it satisfies (NHA 2). The category CGH of compactly generated Hausdorff spaces and continuous mappings becomes our abstract homotopy category and so does the pointed category CGH* of CGH (cf. Example 1.7 in [7]). We say that two morphisms /„, /i: X-*Y are homotopic (relative /: A-*X\ if there is a morphism /: IX-+Y with /*=/&* for fe=0, 1 (and //y=/0;V); and then we write /o-/i (rel /) and call / a homotopy of /0 and /t. When /, g: IX-+Y are homotopies with /l*=gO#, we can define a sum /0g of homotopies / and g as usual which is unique up to homotopy relative /={0*Jil#} (the terminal MAUNDER TYPE THEOREMS 633 faces). Here we note on the dual considerations. By using the unit t]: Id^PI and the counit e: IP-*Id, we have the following axiom (A3*) which is dual and equivalent to (A3) by defining k*—zk*P (k=Q, 1) and a=P(r}r]: (A3*) There are three natural transformations k*:P^Id (&=0, 1) and a: Id-*P with §*a—Id=l*a, called the top-coface, bottom-coface and injection transformations respectively. By using (A3*) and dual constructions, we can obtain the duality principle in our abstract homotopy (cf. [7]). Let C be a pre-homotopy category. We say that /: A-^X in C (resp. p: Y—+B) has the relative HEP (resp. relative HLP), if any commutative square