The surgery obstruction groups for finite 2-groups

Hambleton, Ian; Milgram, R. James

doi:10.1007/bf01389893

Cited by 25 publications

(16 citation statements)

References 16 publications

(17 reference statements)

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“…To see that the lifting of 6v(f) is framed bordant to zero, it suffices to note that the lifting of 6v(f) is homotopic to fly(f2), where f2 represents the corresponding class in n 1 (Autz/4(SS))=Z 4 9 Z2 and that 6v(f2)=0 by part We can now complete the proof of the main result in this section. (2V)) (see [17]) and the oozing problem, we see that the image of y is nonzero for all G if and only if it is nonzero for Q (8). Using methods beyond those of this paper, Cappell and Shaneson have verified that the image of y is in fact nonzero [1 i].…”

Section: Corollary 44 the Unique Nonzero Element U2 Of ~Zl(autq(s3)mentioning

confidence: 85%

“…It is implicit in the statement of the theorem that the relevant Wall group always has order at least 4. In fact,/21(G) is always at least as large as /2~(G2), where G2 is the Sylow 2-subgroup of G, and it is known that if G2 has order 2 p (p> 3) then/21(G2) is a 712-vector space of dimension 2 p-2_p + 3 (compare Wall [551 p. 69, or Hambleton-Milgram [17], p. 34). In particular, the quotient group/2 I(G)/I o has at least two elements if G2 has order > 16, and it follows that nontrivial s-cobordisms with both ends homeomorphic to S3/G exist for all such groups G. If the Sylow 2-subgroup has order 8 then nontrivial s-cobordisms exist in some cases but not in others, and no clear pattern exists at this time.…”

Section: First Classification Theorem Let G Be a Finite Group That Amentioning

confidence: 96%

“…On the other hand, it is known that j. is injective (compare [17]), and therefore it suffices to determine B, when 7t~Qs. Cappell and Shaneson show that B~ is nonzero in this case by a delicate argument involving the visible symmetric L-groups defined by M. Weiss (see [57], Section 1).…”

Section: Comparison With the Quaternionic Casementioning

confidence: 99%

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Ons-cobordisms of metacyclic prism manifolds

Kwasik

Schultz

1989

Invent Math

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In geometric topology it is important to have useful criteria for recognizing special types of manifolds. For example, it is often necessary to determine whether a connected compact manifold with boundary W can be expressed as a product V• [0, 1]. If W is two-dimensional, then the classification of surfaces shows that some obvious necessary conditions are sufficient. Specifically, the boundary must split into two components, and the inclusion of either must be a homotopy For many years it has been known that the s-cobordism theorem extends to certain specific situations in dimension 3, 4, and 5 (compare Lawson [27], Shaneson [48]) but it was also known that the s-cobordism theorem does not generalize completely in at least one of these dimensions (Siebenmann [49], Lawson [27]). In fact, it was known that failures occur in each of the three manifold categories (topological, PL, and smooth manifolds).Recent advances in low-dimensional topology have improved our understanding of the extent to which the s-cobordism theorem can and cannot be generalized. The results of M. Freedman yield a topological s-cobordism theorem in dimension five provided the fundamental group of W is relatively small [16,39]. On the other hand, Freedman's results and earlier work of T. Matumoto and L. Siebenmann [32] (cf. [23]) show that the topological s-cobordism theorem fails in dimension four; one can even insist that both boundary components are homeomorphic (specifically to S 1 x RP2). Subsequent work of S. Cappell and J. Shaneson yielded orientable examples of the same type [9]; specifically, * The second

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