SPLITTING ∑(CP∞ × CP∞) LOCALIZED AT 2

Eccles, Peter J.; Mitchell, William P. R.

doi:10.1093/qmath/39.3.285

Cited by 3 publications

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“…We can determine the Poincare series of the stable summands of SX 2 (co) using a technique of [6]. Adjoining a cube-root of unity a) to Q, we let a = x + coy, b = a>x + y.…”

Section: ° ' 1-li Ojmentioning

confidence: 99%

Stable summands ofU(n)

Crabb

Hubbuck

McCall

1997

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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SynopsisThe special unitary group SU(n) has the stable homotopy type of a wedge of n − 1 finite complexes. The ‘first’ of these complexes is ΣℂPn–1, which is well known to be indecomposable at the prime 2 whether n is finite or infinite. We show that the ‘second’ finite complex is again indecomposable at the prime 2 when n is finite, but splits into a wedge of two pieces when n is infinite.

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“…We can determine the Poincare series of the stable summands of SX 2 (co) using a technique of [6]. Adjoining a cube-root of unity a) to Q, we let a = x + coy, b = a>x + y.…”

Section: ° ' 1-li Ojmentioning

confidence: 99%

Stable summands ofU(n)

Crabb

Hubbuck

McCall

1997

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…As a consequence A 3 7<3> (and so X 3 ) is odd and therefore a power of 3 by Corollary 2.5. The calculation is simplified by localizing at the prime 2 and exploiting the splitting of S(CP°° x CP 00 ) described in [8]. Familiarity with the notation of that paper is assumed but for simplicity we write eeZ i2) M (2,Z) for the idempotent e 2 and A'for the stable 2-local complex 'Z~1X 2 .…”

Section: The Case In Which the Codimension Ismentioning

confidence: 99%

“…A basis for H*(X) is given in [8,Corollary 2.3]. This determines a basis for H*(X A CP 00 ) = H*(X) ® #*(CP°°) and so a dual basis for H+{X A CP 00 ).…”

Section: Proposition F?(ul) =mentioning

confidence: 99%

“…. because c(v t ) = n f + ff ( -2 and ch (%) = e x *.Similarly ch(u\u 2 ) = (e*i-l)8 (e*«-l) = X?JC 2 + . and the formula for ch (M X M 2 ) is given by symmetry.A similar calculation to that for Proposition 4.2 shows that, in ffj^QCP 00 x CP°°), x\ x 2 , 2x\ x\ and x\ x\ survive to E\ 2 0 and that El 3 has generators xj x 2 ® v, ^ x 2 ® v (of order 12) and x\x\ ® v (of order 24).…”

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confidence: 97%

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Triple Points of Immersed Orientable 2n -Manifolds in 3n -Space

Eccles

Mitchell

1989

Journal of the London Mathematical Society

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Boy's surface [4] is a self-transverse immersion of the real projective plane into Euclidean 3-space with a single triple point. J. F. Hughes has observed [10] that for each positive integer n there is a self-transverse immersion of a closed 2w-manifold in Euclidean 3«-space with an odd number of triple points. An example is obtained by taking the cartesian product of n copies of Boy's immersion and making it selftransverse using a regular homotopy. These examples are all non-orientable. In this paper we consider when orientable examples can be found. THEOREM. There is a self-transverse immersion M 2n q'^U Zn of some closed orientable 2n-manifold M with an odd number of triple points if and only ifn is even and n > 4.This is a simple consequence of the more technical Theorem 1.2 stated below. For n = 1 it is well known (see, for example, [2]) that a self-transverse immersion of a closed orientable surface in U 3 has an even number of triple points. Hughes observes that the same is true for orientable 4-manifolds in U 6 .Our results are obtained using the approach of [6] which is based on ideas of U. Koschorke and B. Sanderson [12] (also developed independently by P. Vogel [18] and J . Lannes [13], and by A . Sziics [17]). The method is homotopy theoretic; explicit manifolds and immersions are not constructed.In order to state the main result it is necessary to review some of these ideas and to explain how the number of triple points of an immersion can be viewed as a bordism invariant. Further details may be found in [5,6,7].Given a self-transverse immersion /: M 2 "^ <*+ |R 3n of a closed oriented 2w-manifold, compactness ensures that the number of triple points is finite. If n is even a sign may be attached to each triple point by comparing the standard orientation of IR 3n with that provided by the orientation of the three normal planes at that point. Since the normal planes are unordered this is impossible if n is odd. We write 0 3 (i) for the number of triple points of the immersion, taking the number modulo 2 if n is odd and counting the points in a signed way if n is even.Recall that the Pontrjagin-Thom construction gives an isomorphism from the bordism group of closed oriented 2«-manifolds immersed in IR 3n to the stable homotopy group n$ n (MSO(n)) where MS0(n) is the Thorn complex of a universal

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A group ring package for Cayley

Sandling¹

1991

SIGSAM Bull.

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SPLITTING ∑(CP^∞ × CP^∞) LOCALIZED AT 2

Cited by 3 publications

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Stable summands ofU(n)

Stable summands ofU(n)

Triple Points of Immersed Orientable 2n -Manifolds in 3n -Space

A group ring package for Cayley

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