Papers by Richard Milgram
In this note we analyze the topology of the spaces of con- figurations in the euclidian space R,.
Lecture Notes in Mathematics, 1978
this paper we use geometrically inspired methods to make computations. General results on the L-t... more this paper we use geometrically inspired methods to make computations. General results on the L-theory of a Laurent polynomial extension are used to study the Witt groups of genus 0 function fields. Given a ring A with involution # : a-#a and a central unit u let A[t, t -1 ] u denote the Laurent extension ring A[t, t -1 ] with the involution t = ut -1 , abbreviated to A[t, t -1 ]foru = 1. In Theorem 4.1 we establish the exact sequence ..
Lecture Notes in Mathematics, 1979
Lecture Notes in Mathematics, 1981
ABSTRACT Without Abstract
Lecture Notes in Mathematics, 1985
Lecture Notes in Mathematics, 1987
I. Introduction We have made computer calculations of the E 2 terms of the unstable Adams spectra... more I. Introduction We have made computer calculations of the E 2 terms of the unstable Adams spectral sequences which converge to the homotopy groups of the spheres. We use the A algebra and an algorithm based on EHP sequences to calculate the unstable Adams E 2 term through ...
Proceedings of the London Mathematical Society, 1993
Five of the sporadic simple groups have maximal subgroups involving M 21 = L 3 (4). They are M 22... more Five of the sporadic simple groups have maximal subgroups involving M 21 = L 3 (4). They are M 22 , M23, M C L, HS and O'N. In this paper we study the cohomology rings of these maximal subgroups and several further groups closely associated with L 3 (4). Contents 0. Introduction 187 1. The Tits building for L 3 (4) 190 2. The cohomology of Syl 2 (L 3 (4)) 192 3. The cohomology of 2 4 : jtf 4 , 2 4 : s£ 5 , and L 3 (4) 196 4. The groups 2 4 : sd A : 2 2 , 2 4 : s£ 5 : 2 2 , and L 3 (4) : 2 2 200 5. The ^-invariant subring of F 2 [JC, y, z, w] 208 6. Two extensions connected with the O'Nan group 212 7. The extensions 2 5 : s£ s and 2 • L 3 (4) associated to O'N 217 References 224 Odd index no yes yes no yes 2-Local no no no no yes isomorphism jtf 8 = GL 4 (F 2 ). The next level of difficulty from this point of view is represented by L 3 (4) (note the rather special fact that \s# 8 The cohomology of L 3 (4) seems to have been first studied by Yagita [14]; however, there seem to be some errors in the results there. Using techniques very close to those of [14], David Benson gave a correct calculation of //*(Syl 2 (L 3 (4))) which he was kind enough to show to one of us. We reproduce Benson's and Yagita's calculation in §2. From that point on, however, our approach differs considerably from that of [14]. To analyse the cohomology of L 3 (4), we first decompose it using the Tits building T(L 3 (4)). In this case it is a graph with an edge-transitive action of L 3 (4) and orbit space where the vertex stabilizers P x and P 2 are the usual parabolic subgroups (in this case 2 4 : s& 5 since L 2 (4) = s$ 5 ) and B is the Borel subgroup of upper triangular matrices (2 4 : s£ 4 in this case). From the well-known fact that H X (T(L 3 (4))) is a projective F 2 (L 3 (4)) module-indeed it is the Steinberg module St 3 (4)-we obtain the cohomological decomposition (see also [13]) H*(L 3 ( )) 0 H*(B) = H*(P l )®H*(P 2 ). This type of decomposition can be modified to deal with extensions of L 3 (4) as well. The groups P x , P 2 are isomorphic, and, as has been noted, are given as semidirect products In § 1 we show how H*(P) decomposes via the formula H*{P) 0 H*((I/2)Y 4 = //*(Syl 2 (L 3 (4))) z/3 0 // Using this, the calculation is reduced to computing H*(B) ( § 3) and to calculating the rings of invariants ( § 5). Next we turn to L 3 (4): 2 2 and its subgroups 2 4 : S^ and 2 4 : 5^5. We first determine //*(Syl 2 (L 3 (4): 2 2 )) and then, using double coset decompositions, we work our way from H*(2 4 : ^4) to H*(2 4 : Sf 5 ) and finally to H*(L 3 (4): 2 2 ). We point out that Syl 2 (L 3 (4):2 2 )^Syl 2 (M 22 ); We proceed to determine ( S= F 2 [v, w, a, b](l, R, S, RS) {F) . Note that F + 1 is a derivation. Consequently, we filter ^ by F 2 [t>, w, a, b] c F 2 [u, w, a, b](l, R, S) <= ôbtaining Ker(F + 1): F 2 [u, w, a, b](R, S)-> F 2 [u, w, a, b] = F 2 [u, w, a, b]{aR + bS), and F + 1: F 2 [u, w, a, b]RS^> F 2 [v, w, a, b](R, S) is an injection since (F + 1)(RS) = (a + b){aR + bS).
Mathematische Zeitschrift, 1993
In the authors gave a description of the stable homotopy type of the space of degree k basepoint ... more In the authors gave a description of the stable homotopy type of the space of degree k basepoint preserving holomorphic functions from the Riemann sphere S 2 to the complex projective space II21P". We denoted this space by Ratk(C~" ). Let Dj = F (N. 2, k) + AZk S k be the k th subquotient of the May-Milgram model for ~2S3 . Here F(~ 2, k) is the configuration space of k-disjoint points in R E and ~k is the symmetric group on k letters which acts on both FOR E, k) and the sphere S k =S 1 ^ .../x S 1 by permuting coordinates. Then one of the main results of [3] is
Journal of Group Theory, 2000
and, as a result of our calculation Hi(M23;Z) = 0 for i < 5. In particular M23 is the first kn... more and, as a result of our calculation Hi(M23;Z) = 0 for i < 5. In particular M23 is the first known counterexample to the conjecture that if G is a finite group with Hi(G;Z) = 0, i = 1, 2, 3, then G = {1}‡. (M11, the first Mathieu group and J1, the first Janko group also satisfy Out(G) = Mult(G) = 1, but for both of these groups H3(G;Z) = 0.) It would be tempting to amend the conjecture. It is very likely that it only fails for a very small number of the sporadics among the simple groups. So one might well suspect that there is a (small) finite number n so that H(G;Z) = 0 for 0 < i ≤ n implies that G = {1} if G is finite. But I have no idea as to a suitable candidate for n.
Forum Mathematicum, 1990
For any 2-group π, two homomorphisms from H2(n) to Lo(ZM) one defined purely algebraically and on... more For any 2-group π, two homomorphisms from H2(n) to Lo(ZM) one defined purely algebraically and one defined via surgery are shown to be equal. 1980 Mathematics Subject Classification (1985 Revision): 19G24, 57R67, 19B28. For any group π, a homomorphism K2 : H2(n) * Q2(Bn) -» LJ is defined geometrically s follows. Fix any element χ e H2 (n) = Ω2 (Bn), and represent it by a map/: M -> Bn, where M is some oriented surface. Then κ2 (χ) is the surgery obstruction (over Ζ[π]) for the product of IdM with the Kervaire problem The map K2 s just defined is actually (by abuse of notation) the restriction of a homomorphism κ2 : Η2 (π; Z/2) -» Lg (Ζ[π]), used in [5] when studying the surgery obstructions for normal maps between closed manifolds. In fact, κη is defined for all n; and for finite π the image of the composite //„(π; Ζ/2) is described in [5, Theorems 6.8 and 5.4]. Here, Ζ^(Ζ[π]) is the surgery obstruction group for constructing homotopy equivalences with torsion in SKi(Z[nJ). A complete understanding of κη itself thus depends on understanding SKl (Ζ[π]) and its interaction with surgery obstructions, and seems out of reach for now. However, Standard results [l 6] reduce the problem (for finite π) to the case where π is a 2-group. For any finite π, a subgroup Cl^ (Ζ[π]) s S^ (Ζ[π]) is defined so that the sequence 52 R. J. Milgram, R. Oliver is exact [10, Theorem 3.15]. In order to understand K2 s a map to Lg(Z[n])y we study, for a 2-group π, the composite The main result in this paper is a purely algebraic description of / ° /c2, first conjectured by Hambleton, and based on the isomorphism 2M) — ^(02M)] — Η2(π)/Η?(π) constructed in [9] (see also [10, Chapter 8]). Here, H2(n) is the subgroup ofH2(n) generated by the images of H2(o) for abelian ρ ̂ π. The Standard involution on SKV (Ϊ2 M) is seen in [10, Theorem 8.6] to be given by multiplication by — l . Thus, Θ ~ 1 induces a natural surjection Ω : H2 (π) — » SK^ (22 [π]) (χ) Ζ/2 * ft (Z/2; 5^ (*2 [π])) ; where ft (Z/2; -) denotes the T te cohomology group with respect to the Standard involution. The composite of Ω with the "hyperbolic" map h : 1 (Z/2; S K, (22 [π])) -> Lg (^ [π]) in the Rothenberg exact sequence now provides the second homomorphism from H2(n) to Lo(/2M). The commutativity of the square
Copyright © 1979 by Princeton University Press ALL RIGHTS RESERVED Published in Japan exclusively... more Copyright © 1979 by Princeton University Press ALL RIGHTS RESERVED Published in Japan exclusively by University of Tokyo Press in other parts of the world by Princeton University Press Printed in the United States of America by Princeton University Press, Princeton, New ...
The Annals of Mathematics, 1970
... In the category of the nth loop spaces and loop maps there are homology operations over and a... more ... In the category of the nth loop spaces and loop maps there are homology operations over and above those which are obtained as the homology duals of ... W. Browder then classified the operations on more than one variable in his thesis and also studied some mod p operations. ...
ABSTRACT In this paper we develop a systematic topological approach to motion planning for a plan... more ABSTRACT In this paper we develop a systematic topological approach to motion planning for a planar 2-R manipulator with point obstacles. By considering components in the free space for the second joint as the first joint varies, we build a two-dimensional array representing the cells of the free space and an asociated graph representing the boundaries of those cells. Using this graph, we derive a closed formula for the number of components of the free space. At the same time we solve the motion existence problem, namely, when are two arbitrary configurations in the same component? If so, we develop two explicit algoritms for constructing the path -a middle path method and a linear interpolation method. These algorithms give complete solutions to the path planning problem. Extensive examples are worked out which verify the correctness and efficiency of the resulting program. Then we briefly discuss how these methods generalize to a 3-R planar manipulator.
Algebraic and Geometric Topology, 1978
A process for preparing allylic esters of carboxylic acids and allylic alcohols which comprises r... more A process for preparing allylic esters of carboxylic acids and allylic alcohols which comprises reacting an olefin having an allylic carbon-hydrogen bond, a lower alkyl carboxylate ester, water and oxygen in the presence of a catalyst system comprising an oxidation catalyst and an acidic co-catalyst.
Proceedings 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems. Expanding the Societal Role of Robotics in the the Next Millennium (Cat. No.01CH37180)
Maximizing the use of dual-arm robotic systems requires the development of planning algorithms an... more Maximizing the use of dual-arm robotic systems requires the development of planning algorithms analogous to those available for single-arm operations. In this paper, the global properties of the configuration spaces of planar n-bar mechanisms (i.e., kinematic chains forming a single closed loop) are used to design a complete motion planning algorithm. Numerical experiments demonstrate the algorithm's superiority over a typical algorithm that uses only local geometric information.
Asian Journal of Mathematics, 1997
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Papers by Richard Milgram