The Kervaire Invariant of Framed Manifolds and its Generalization

Browder, William

doi:10.2307/1970686

Cited by 220 publications

(199 citation statements)

References 15 publications

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“…Investigating this modified Machine (3) with G = {1}, one finds that it reproduces more or less the "generalized Kervaire invariants" of [1,2] itself as "the" generalized Kervaire invariant: it requires no choices or restrictions of any kind and, more important, it gives very slick product formulae. The computation of the groups L 2k (B, 7) is easy in the case at hand (use the theorem, and bear in mind that any chain complex over Z 2 is homotopy equivalent to its homology).…”

Section: Here a Chain Bundle On A Chain Complex D Is A O-dimensional mentioning

confidence: 97%

Surgery and bordism invariants

Weiss¹

1983

Bull. Amer. Math. Soc.

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Section: Here a Chain Bundle On A Chain Complex D Is A O-dimensional mentioning

confidence: 97%

Surgery and bordism invariants

Weiss¹

1983

Bull. Amer. Math. Soc.

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“…In [3] Browder developed α : Ω 2n (ξ) −→ Z 2 , n even or odd, ξ −→ X, as follows: One may assume that X is a smooth closed N -manifold, N large, with normal bundle in R N +k equal to ξ. Suppose M is a smooth closed 2n-manifold with ξ-structure f .…”

Section: Surgerymentioning

confidence: 99%

“…As an expository device we describe the development of this subject in chronological order beginning with Kervaire's original paper ( [10]) and Kervaire-Milnor's Groups of Homotopy Spheres ( [11]) followed by Frank Peterson's and my work using Spin Cobordism ( [5], [7]), Browder's application of the Adams spectral sequence to the Kervaire invariant one problem ( [3]), Browder-Novikov surgery ( [16]) and finally an overall generalization of mine ( [6]). In a final section we describe, with no detail, other work and references for these areas.…”

Section: Introductionmentioning

confidence: 99%

The Kervaire invariant and surgery theory

Brown¹

2000

Princeton University Press eBook Package 2014

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“…+ will be well defined once we show that a'i = ai. Define H": Sn + kx [2, A] -> F(^) by H"ix,u) = H\x, -u + 6) for we [2,3] and H"ix, u) = Hix, u) for u e [3, A]. Let W" = iH"Y\M) and h" = H"\ W".…”

Section: Borhood Of V In W T(vf) = E/de Is the Thorn Complex Of Vt mentioning

confidence: 99%

Groups of embedded manifolds

Agoston¹

1971

Trans. Amer. Math. Soc.

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Abstract. This paper defines a group B(Mn, vv) which generalizes the group of framed homotopy zz-spheres in Sn*k. Let M" be an arbitrary 1-connected manifold satisfying a weak condition on its homology in the middle dimension and let v" be the normal bundle of some imbedding ■ T(v0) is a map which is transverse regular on M, V = F~1(Mn), and f=F\V is a homotopy equivalence, (r^) is the Thorn complex of v0.) There is a natural group structure on 0(M", vv), and 6(Mn, v") fits into an exact sequence similar to that for the framed homotopy zz-spheres.This paper attempts to generalize in a natural way a well-known exact sequence concerning framed homotopy spheres which is contained in the work of Novikov [11], , and Levine [10]. The author stumbled onto these results partly because of his efforts to prove imbedding theorems for manifolds in the metastable range, and partly because of his recent work on Browder-Novikov theory for maps of degree d, \d\^0 (see [2]).§2 describes the basic constructions used in this paper. The "group of embedded manifolds", 8(M, v0), is defined in §3. A fairly simple description ofthat group is given towards the end of that section. §3 also contains the main results about 8(M, Vq). In §4 we discuss a few interesting open problems. The author would like to thank the referee for some helpful suggestions.1. Notation. All manifolds will be C", compact, and oriented. Maps will be transverse to boundaries.If Mn is a connected closed manifold, let [M]eHnM denote the orientation class. Recall that/: Vn-^Mn is said to have degree d, i.e., deg/=c7, iff*([V}) = d [M], where/*: HnV^-HnM is the map induced by/on the integral homology groups.As usual, Dk denotes the closed unit ball in Euclidean zc-space Rk, i.e., Dk = {(yx,...,yk)eRk\y2+---+yl^l}. Sk = 8Dk + 1 = Dk+ U Dk_, where D% ={(yi,---,yk + i)eRk + 1\y2x+---+y2k+x = l,yaO} and Dk.={(yx,..., yk + x) eRk+1 \y\+---+yl + x = l,yxúO}. We have natural inclusions Sk^Sk+1 and DkçDk + 1. Lete = (l,0)eS°^Sk.

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The Kervaire Invariant of Framed Manifolds and its Generalization

Cited by 220 publications

References 15 publications

Surgery and bordism invariants

Surgery and bordism invariants

The Kervaire invariant and surgery theory

Groups of embedded manifolds

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