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## Surgery and the generalized Kervaire invariant (1985)

Venue: | Universitdt Bielefeld |

Citations: | 13 - 1 self |

### Citations

257 | Surgery on compact manifolds,
- Wall
- 1970
(Show Context)
Citation Context ...lgebraic theory of surgery'; see [15]. Secondly, it is computationally satisfying. Thirdly, it has applications to classical surgery theory, especially to the calculation of the symmetric L-groups of =-=[13]-=- and [15]; and therefore to anything which involves product formulae for surgery obstructions. A black box description of the theory has been given in [22]; in this introduction I shall concentrate on... |

84 | The algebraic theory of surgery.
- Ranicki
- 1980
(Show Context)
Citation Context ...undle of I, 3.4, exterior Whitney sums (explanation below) correspond to exterior products of chain bundles. Proof. The passage from spherical fibrations to chain bundles in I, 3.4 (and previously in =-=[11]-=-) was based on equivariant S-duality; 9.3 is an application of the principle that (equivariant) S-duality commutes with smash products. Details are again left to the reader. To get a good definition o... |

83 |
Exact sequences in the algebraic theory of surgery. In:
- Ranicki
- 1981
(Show Context)
Citation Context ...MICHAEL WEISS [Received 1 November 1982—Revised 24 January 1984] Introduction (i) Synopsis. The discovery, around 1960, of the 'Kervaire Invariant' for almost framed manifolds of dimension 4k+ 2 (see =-=[12]-=-) was an important stimulant for the development of surgery theory; but it also led to the theory of the 'generalized Kervaire Invariant' of Browder and Brown [2, 3]. The present paper is an attempt a... |

59 |
Immersions of manifolds, Trans
- Hirsch
- 1959
(Show Context)
Citation Context ...the same domain and range. 5388.3.51 K 162 MICHAEL WEISS There are at least two distinct ways (1.9 and 1.11) of making the chain bundles over a fixed chain complex C in (€A into a simplicial set (see =-=[6]-=- for general information on simplicial sets). 1.9. DEFINITION. The 'Kan-Dold simplicial set of chain bundles on C is the simplicial set KD{W&C~*) obtained by applying the Kan-Dold functor KD to the ch... |

58 |
The Kervaire invariant of framed manifolds and its generalization,
- Browder
- 1969
(Show Context)
Citation Context ...3 ence ^(n, w; X, y; a, j) ~ (Z,0). But the following argument is easier. Pick another admissible 0-cycle s' in ^(n, w; X, y; a, j). Crossing the string of data (n,w;X,y; a, j) with the unit interval =-=[0,1]-=- gives a new string, written (n,w;Xx[0, l ] , yx [0 , l ] ; a , ; ) . Choose an admissible 0-cycle s" in 0>(n, w; X x [0, l],y x [0, l ] ; a , j) whose image under the restriction map -• 0>{n, w;Xx {0... |

36 |
Generalizations of the Kervaire invariant
- Brown
- 1972
(Show Context)
Citation Context ..., v2, w 4 , . . . ) . This completes the proof. 11. Miscellany This section contains two distinct illustrations of the theory. The first is related to the 'generalized Kervaire invariants' of [1] and =-=[2]-=-, and the second is a not-so-new proof of Browder's theorem [1] on the Kervaire invariant (which sheds light on the results of § 10, but not on Browder's theorem). We shall work with CW-spaces instead... |

33 |
Pr iddy, Applications of the transfer to stable homotopy theory ,
- Kahn, B
- 1972
(Show Context)
Citation Context ...icial set X with a simplicial 7i-action which freely permutes the simplices of X, and an identification of simplicial sets X/n = X. Suppose that X and Y are simplicial sets. The acyclic model theorem =-=[9]-=- yields a chain homotopy equivalence C(XxY) - C{X)®ZC{Y) natural in both variables with respect to simplicial maps. (Note: we are talking about cellular chain complexes.) To be more thorough, the acyc... |

22 |
Surgery on Poincaré and normal spaces
- Quinn
- 1972
(Show Context)
Citation Context ...re r > 0, T being the generator of Z2), and write W&C for the abelian group chain complex HomZ[Z2](W,C®AC). Then Q"(C):=Hn(W&C) is the ( — n)th cohomology group of Z2 with coefficients in C ®AC. (See =-=[10]-=- if the terminology appears mysterious.) On replacing the standard resolution W by the standard complete resolution W (with Wr = Z[Z2] for all r, d: Wr -• ^ . , ; X H ( 1 +(-)rT)-x, for all r) we obta... |

11 |
Halbexakte Homotopiefunktoren
- Dold
- 1966
(Show Context)
Citation Context ...n.) For C in Q)A, let (Here C$k is the /c-skeleton of C.) So the homotopy-invariant functor (i) CH+^<2° (C - * ) is the '/cth Postnikov base' of the (homotopy-invariant) functor (ii) Ch+<2°(C-*); see =-=[4]-=- for a completely analogous topological definition. Or in other words, passing from (ii) to (i) amounts to 'killing' the coefficient groups of the functor (ii) in dimension greater than k, that is, th... |

8 |
La classification des immersions combinatoires
- Haefliger, Poenaru
- 1964
(Show Context)
Citation Context ...sociated algebraic bordism spectra J?(B, 6) and l°(B, 6). (5£\B,6) is a more sophisticated version of l°(B, 6), with better algebraic properties.) Inspiration comes from the mock-bundle philosophy of =-=[5]-=-. 2.1. DEFINITION [13,15]. An ln dimensional algebraic Poincare complex (over A)y is a pair (C, q>) consisting of a positive chain complex C in (€A (that is, Cr = 0 for r < 0) and an n-dimensional cyc... |

4 |
Kervaire’s invariant for framed manifolds
- JONES, REES
- 1978
(Show Context)
Citation Context ...ne obtained in this way (from a suitable chain complex B). This is the analogue in the chain complex world of E. H. Brown's representation theorem, which normally lives in the world of CW-spaces; see =-=[7]-=-. We shall now prove it in detail for the special case of the cohomology theory If B is an arbitrary chain complex of projective /4-modules (not necessarily in <€A), the abelian group chain complex W&... |

3 | A note on the Kervaire invariant
- Jones, Rees
- 1975
(Show Context)
Citation Context ...logy of [6], EG is a twisted cartesian product with base BG and fibre G.) Let p: EG —> BG be the projection (g, b) \—* b. If no(G) is a group, then the geometric realization of p is a quasi-fibration =-=[8]-=-. If moreover G is a Kan simplicial set, then so is BG (this is proved in 3.21 below). Under these conditions it follows easily that nn{BG) = ^ . ^ G ) for all n, and that EG is contractible. (Even so... |

1 |
A remark concerning immersions of S" in R2/", Quart
- BROWN
(Show Context)
Citation Context ...ed manifolds of dimension 4k+ 2 (see [12]) was an important stimulant for the development of surgery theory; but it also led to the theory of the 'generalized Kervaire Invariant' of Browder and Brown =-=[2, 3]-=-. The present paper is an attempt at uniting these two theories, by constructing a non-simply-connected and in other respects updated version of the generalized Kervaire Invariant. The construction ha... |