## On the topological Hochschild homology of bu. I. (1993)

Venue: | AMER. J. MATH |

Citations: | 17 - 0 self |

### BibTeX

@ARTICLE{McClure93onthe,

author = {J. E. McClure and R. E. Staffeldt},

title = {On the topological Hochschild homology of bu. I.},

journal = { AMER. J. MATH},

year = {1993},

volume = {115}

}

### OpenURL

### Abstract

The purpose of this paper and its sequel is to determine the homotopy groups of the spectrum THH(l). Here p is an odd prime, l is the Adams summand of p-local connective K-theory (see for example [25]) and THH is the topological Hochschild homology

### Citations

481 | Homological Algebra - Cartan, Eilenberg - 1956 |

261 | Simplicial objects in algebraic topology - May - 1992 |

258 | The geometry of iterated loop spaces - May - 1972 |

203 |
Complex cobordism and stable homotopy groups of spheres
- Ravenel
- 1986
(Show Context)
Citation Context ...Sigma N) �� = Ext \Sigma (Z=p; N) (7) (see [18, Theorem A1.3.12]). Here \Gamma denotes a Hopf algebra, \Sigma a quotient Hopf algebra, and N a \Sigma-comodule; the 2-product is defined on page 311=-= of [18]-=-. (See pages 337--339 of [1] for a dual version of the following argument which avoids the 2-product). First we observe that if N is actually a \Gamma-comodule (more precisely if the \Sigma-comodule s... |

138 | History of the Theory of Number - Dickson - 1952 |

52 |
Cyclic homology, derivations, and the free loopspace, Topology 24
- Goodwillie
- 1985
(Show Context)
Citation Context ... is not precise, however, and in particular if the analog of the Hochschild construction is applied to a topological group G the result is not BG but instead is the free loop space Map(S 1 ; BG); see =-=[11]). 4 Cal-=-culation of the E 2 -term of the Adams spectral sequence We remind the reader that p denotes an odd prime. Let M denote the Moore spectrum S 0 [ p e 1 . By definition we have ��s(X; Z=p) = ��s... |

47 | Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen - Kummer |

41 |
Topological Hochschild homology
- Bökstedt
- 1990
(Show Context)
Citation Context ...trum THH(`). Here p is an odd prime, ` is the Adams summand of p-local connectivesK-theory (see for example [25]) and THH is the topological Hochschild homology construction introduced by Bokstedt in =-=[3]-=-. In the present paper we will determine the mod p homotopy groups of THH(`) and also the integral homotopy groups of THH(L) (where L denotes the periodic Adams summand). In the sequel we will investi... |

41 | Algebraic K-theory of spaces - Waldhausen - 1985 |

31 |
Stable Homotopy and Generalized Homology
- Adams
- 1974
(Show Context)
Citation Context ...; N) (7) (see [18, Theorem A1.3.12]). Here \Gamma denotes a Hopf algebra, \Sigma a quotient Hopf algebra, and N a \Sigma-comodule; the 2-product is defined on page 311 of [18]. (See pages 337--339 of =-=[1]-=- for a dual version of the following argument which avoids the 2-product). First we observe that if N is actually a \Gamma-comodule (more precisely if the \Sigma-comodule structure on N is induced by ... |

22 | Algebraic K-theory of spaces – a manifold approach, in Current trends in algebraic topology - Waldhausen - 1981 |

12 |
Derived tensor products in stable homotopy theory, Topology 22
- Robinson
- 1983
(Show Context)
Citation Context ...e conclude this section with some remarks which will not be used in the rest of the paper. Remark 3.4 Alan Robinson has defined a "topological" analog of Tor S (M; N ), which he denotes by E=-= R F (see [19]-=-). In analogy with the equation HHsS = Tor S\Omega S op (S; S) one presumably has THH(R) ' R RR op R: Remark 3.5 There is another way to relate Robinson's work to THH. Given a sufficiently good map of... |

12 |
Divisibility of Binomial and Multinomial Coefficients by Primes and Prime Powers
- Singmaster
(Show Context)
Citation Context ...j p j \Gamma i !/ s \Gamma p j + i s \Gamma p j ! v p j \Gammai 1 f s\Gammap j +i : But the coefficient of f s has nonzero reduction mod-p, since it is a p-adic unit by [13, page 115], [9, page 270], =-=[20]-=-, or by hand, so induction on s proves the assertion. Thus we have a surjection m : Z=p [v 1 ; v \Gamma1 1 ]\Omega Z=p [f 1 ; f p ; . . .] \Gamma! K(1)s(`): Now we determine the relations satisfied by... |

11 | Multiplicative structures in mod q cohomology theories - Araki, Toda - 1965 |

9 |
Topological Hochschild homology of Z and Z/p
- Bökstedt
(Show Context)
Citation Context ...ory functor, and it is very accessible to calculation. We shall review what is known about the first property in a moment; the second property was demonstrated by Bokstedt's calculation, in his paper =-=[4]-=-, of the homotopy groups of THH(HZ=p) and THH(HZ) (here HZ=p and HZ denote the evident Eilenberg-Mac Lane spectra). It is natural to ask about THH(R) for other popular ring spectra R, and our work is ... |

8 |
Algebraic K-theory of spaces, localization, and the chromatic filtration of stable homotopy.” Algebraic topology, Aarhus 1982
- Waldhausen
- 1984
(Show Context)
Citation Context ...ains the statement we made earlier that connective spectra are of particular importance for the potential applications. We conclude with one further remark about the potential applications of THH. In =-=[23]-=-, Waldhausen has proposed an interesting program for studying the relative K W theory of the map S 0 ! HZ by means of the intermediate spectra K W (L n (S 0 )) and K W (L n (S 0 ) c ), where L n (S 0 ... |

5 |
Morava stabilizer algebras and localization
- Miller, Ravenel
- 1977
(Show Context)
Citation Context ... 0 and m can be unwound to give the rest of the proof. --- Our second proof of Proposition 5.4 follows a suggestion of M.J. Hopkins that a proof could be extracted from a result of Miller and Ravenel =-=[17]-=- in stable homotopy theory. We follow the exposition in [18] between pages 223 and 226. Obviously the prerequisites for this proof are much more demanding than those for the first proof, but we can al... |

3 |
On the topological Hochschild homology of
- McClure, Staffeldt
- 1993
(Show Context)
Citation Context ...s is what topological Hochschild homology THH(R) is. It is clear enough in principle how one should construct THH(R) (see Section 3), although the technical details are quite complicated (see [3] and =-=[10]). THH(R-=-) is a spectrum and we shall denote its homotopy ��sTHH(R) by THHs(R). There is a natural transformation �� 0 : K W R ! THHsR which is analogous to the Dennis trace map. (See [3, Section 2] fo... |

2 | The map BSG - Bökstedt, Waldhausen - 1987 |

1 |
The natural transformation from K(Z) to THH(Z
- Bokstedt
- 1987
(Show Context)
Citation Context ...tedt has shown that �� 0 is nonzero in infinitely many dimensions; more precisely, what he shows is that for each prime p the localization of �� 0 at p is an epimorphism in dimension 2p \Gamma=-= 1 (see [5]). N-=-ote that this cannot be true for �� for the trivial reason that HHsZ is zero in all positive dimensions. In the cases R = S 0 and R = \Sigma 1(\Omega X)+ mentioned above one can give explicit desc... |

1 | The homotopy theory of H1 ring spectra - Bruner - 1986 |

1 |
Homology operations for H1 and H n ring spectra
- Steinberger
- 1986
(Show Context)
Citation Context ...perations. As we have seen in the previous section, THH(`) is an E1 ring spectrum, and so its homology supports Dyer-Lashof operations Q i : H n (THH(`); Z=p)) ! H n+2i(p\Gamma1) (THH(`); Z=p)); (see =-=[21]). If x is an el-=-ement of dimension 2s then Q s x = x p , ([21, Theorem 1.1(4)]) so in particular we have (~oes(\Sigma(���� i ))) p = Q p i ~ oes(\Sigma(���� i )): But Bokstedt shows that the map ~ oes... |