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Iterating the cobar construction. (English) Zbl 0843.55002

Mem. Am. Math. Soc. 524, 141 p. (1994).
Since [D. Quillen, Ann. Math., II. Ser. 90, 205-295 (1969; Zbl 0191.53702)] is known, that the DG-coalgebra structure on the chain complex \(C_*(X)\) is a complete homotopy invariant for simply connected rational spaces. Searching complete invariants outside of rational category seems one should enrich the algebraic structure of \(C_*(X)\), at last take into account the existing additional structures such as duals for Steenrod’s \(\cup_i\)-products.
Step by step approaching to complete invariant is related to the following problem: to enrich the algebraic structure of \(C_*(X)\) so that it will determine the invariants of iterated loop spaces \(\Omega^n X\).
The classical cobar construction of Adams \(FC_*(X)\) determines \(C_*(\Omega X)\) just as chain complex but not as DG-coalgebra. L. G. Khelaya [Soobshch. Akad. Nauk Gruz. SSR 96, 529-532 (1979; Zbl 0423.55002); see also Tr. Tbilis. Mat. Razmadze 83, 102-115 (1986; Zbl 0616.55011)], using duals for Steenrod’s \(\cup_i\)-products, has introduced on \(C_*(X)\) the additional structure, which determines in \(FC_*(X)\) a coproduct which is geometric, i.e. corresponds to coproduct of \(C_*(\Omega X)\), but it is just homotopy associative, and the additional structure itself is lost, thus there is no possibility to produce the next cobar construction \(FFC_*(X)\).
The strictly associative geometric coproduct on \(FC_*(X)\) was constructed by H. J. Baues [Compos. Math. 43, 331-341 (1981; Zbl 0478.57027)], thus the author is able to use this coproduct to iterate the cobar construction once but no more because “it is impossible to construct a ‘nice’ diagonal on \(FFC_*(X)\)”.
V.A. Smirnov [Math. USSR, Izv. 27, 575-592 (1986); translation from Izv. Akad. Nauk SSSR, Ser. Math. 49, No. 6, 1302-1321 (1985; Zbl 0595.55008)] has introduced in \(C_*(X)\) an additional structure – the structure of coalgebra over a suitable operad \(E\). This \(E\)-coalgebra structure is transferable on the modified cobar construction of \(C_*(X)\), thus it is good for iteration. Besides, it determines the weak homotopy type of \(X\).
In the present paper the author introduces the notion of \(m\)-coalgebra, and constructs this structure on \(C_*(X)\). The structure of \(m\)-coalgebra consists of cooperations of type \(C_*(X) \to \otimes^n C_*(X)\) satisfying some coherence conditions. The structure involves the usual coproduct and Steenrod’s \(\cup_i\)-products. It determines on \(FC_* (X)\) the geometric coproduct, which is not necessarily associative, but it is a part of certain \(A(\infty)\)-coalgebra structure. The last fact allows to produce the next cobar construction \(FFC_*(X)\). The key point is that \(FC_*(X)\) itself is equipped with the structure of \(m\)-coalgebra, which contains the mentioned \(A(\infty)\)-coalgebra structure, and which is geometric, too: it corresponds to the \(m\)-coalgebra structure of \(C_* (\Omega X)\). This guarantees, under suitable connectedness assumptions, the possibility of the iteration.
The author emphasizes the following differences between Smirnov’s and his own structures: Smirnov’s structure is uncountably generated in all dimensions and there is no obvious connection between those operations and more commonly used operations such as Steenrod’s; Smirnov uses some modification of the cobar construction while the author uses the conventional one. It is mentioned also, that Smirnov’s structure may be more powerful than the author’s: the last structure determines the weak homotopy type for simply connected pointed space, these restrictions do not exist for Smirnov’s structure.
Besides of iteration of the cobar construction the author gives the application of his \(m\)-coalgebra structure to the computation of invariants of total space of fibration. The main tool for the computing of the chain complex of the total space is Brown’s theory of twisted tensor products. But this construction does not give information about the coalgebra structure. The problem is handled in the rational case in works of D. Tanré [Homotopie rationnelle: modèles de Chen, Quillen, Sullivan, Lect. Notes Math. 1025 (1983; Zbl 0539.55001)], N. Berikashvili [Soobshch. Akad. Nauk Gruz. SSR 139, No. 3, 465-468 (1990; Zbl 0741.55012)], S. Saneblidze [Manuscr. Math. 76, 111-136 (1992; Zbl 0766.55008)]. In the integral case the geometric multiplicative structure (non associative) in the twisted tensor product was introduced by M. V. Mikiashvili [Tr. Tbilis. Mat. Inst. Razmadze 83, 46-59 (1986; Zbl 0616.55012)], using the structure of Khelaya, mentioned above.
In the present paper the author computes the \(M\)-coalgebra structure of the integral chain complex of the total space of a fibration. First he computes the \(m\)-structure on the universal acyclic twisted tensor product \(C \otimes_t FC\) and shows that the geometric \(m\)-structure of an arbitrary twisted tensor product \(C \otimes_\xi F\) can be regarded as being induced by that of \(C \otimes_t FC\).

MSC:

55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
55S20 Secondary and higher cohomology operations in algebraic topology
57T30 Bar and cobar constructions
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