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28
Deo/nitions: operads, algebras and modules
 Contemporary Mathematics 202
, 1997
"... There are many different types of algebra: associative, associative and commutative, Lie, Poisson, etc., etc. Each comes with an appropriate notion of a module. As is becoming more and more important in a variety of fields, it is often necessary to deal with algebras and modules of these sorts “up t ..."
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Cited by 25 (2 self)
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There are many different types of algebra: associative, associative and commutative, Lie, Poisson, etc., etc. Each comes with an appropriate notion of a module. As is becoming more and more important in a variety of fields, it is often necessary to deal with algebras and modules of these sorts “up to homotopy”. I shall give a very partial overview, concentrating on algebra, but saying a little about the original use of operads in topology. The development of abstract frameworks in which to study such algebras has a long history. As this conference attests, it now seems to be widely accepted that, for many purposes, the most convenient setting is that given by operads and their actions. While the notion was first written up in a purely topological framework [19], it was thoroughly understood by 1971 [12] that the basic definitions apply equally well in any underlying symmetric monoidal ( = tensor) category. The definitions and ideas had many precursors. I will indicate those that I was aware of at the time. • Algebraists such as Kaplansky, Herstein, and Jacobson systematically studied
Homological stability for the mapping class groups of nonorientable surfaces
, 2007
"... We prove that the homology of the mapping class groups of nonorientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of nonorientable surfaces, up to homology is ..."
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Cited by 18 (4 self)
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We prove that the homology of the mapping class groups of nonorientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that the classifying space of the stable mapping class group of nonorientable surfaces, up to homology isomorphism, is the infinite loop space of a Thom spectrum built from the canonical bundle over the Grassmannians of 2planes in R n+2. In particular, we show that the stable rational cohomology is a polynomial algebra on generators in degrees 4i—this is the nonoriented analogue of the Mumford conjecture.
Determining the Characteristic Numbers of SelfIntersection Manifolds
"... The bordism class of a selftransverse immersion f : M n\Gammak # R n corresponds to an element ff of the homotopy group n\Omega 1 \Sigma 1 MO(k). We explain how the Z=2 Hurewicz image h(ff) 2 Hn(\Omega 1 \Sigma 1 MO(k);Z=2) may be used to determine the characteristic numbers of the se ..."
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Cited by 7 (5 self)
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The bordism class of a selftransverse immersion f : M n\Gammak # R n corresponds to an element ff of the homotopy group n\Omega 1 \Sigma 1 MO(k). We explain how the Z=2 Hurewicz image h(ff) 2 Hn(\Omega 1 \Sigma 1 MO(k);Z=2) may be used to determine the characteristic numbers of the selfintersection manifolds \Delta r (f) of the immersion f . 1 Introduction Given a selftransverse immersion f : M n\Gammak # R n of a compact closed smooth manifold M of dimension n \Gamma k in ndimensional Euclidean space and a positive integer r, the rfold intersection set I r (f) is defined as follows: I r (f) = f f(x 1 ) = f(x 2 ) = : : : = f(x r ) j x i 2 M; i 6= j ) x i 6= x j g: The selftransversality of f implies that this subset of R n is itself the image of an immersion (not necessarily selftransverse) ` r (f ): \Delta r (f) # R n of a manifold \Delta r (f) of dimension n \Gamma kr called the rfold selfintersection manifold of f . (This means in particular that fo...
Categories of spectra and infinite loop spaces
 Lecture Notes in Mathematics Vol. 99, SpringerVerlag
, 1969
"... I presented a calculation of H,(F;Zp) as an algebra, for odd primes p, where F = lim F(n) and F(n) is the topological monoid of homotopy equivalences of an nsphere. This computation was meant as a preliminary step towards the computation of H*(BF;Zp). Since then, I have calculated H*(BF;Zp), for a ..."
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Cited by 6 (4 self)
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I presented a calculation of H,(F;Zp) as an algebra, for odd primes p, where F = lim F(n) and F(n) is the topological monoid of homotopy equivalences of an nsphere. This computation was meant as a preliminary step towards the computation of H*(BF;Zp). Since then, I have calculated H*(BF;Zp), for all primes p, as a Hopf algebra over the Steenrod and DyerLashof algebras. The calculation, while not difficult, is somewhat lengthy, and I was not able to write up a coherent presentation in time for inclusion in these proceedings. The computation required a systematic study of homology operations on nfold and infinite loop spaces. As a result of this study, I have also been able to compute H,(2nsnx;Zp), as a Hopf algebra over the Steenrod algebra, for all connected spaces X and prime numbers p. This result, which generalizes those of Dyer and Lashof [3] and Milgram [8], yields explicit descriptions of both H,(~nsnx;Zp) and H,(QX;Zp), QX = li~> 2nsnx, as functors of H,(X;Zp). An essential first step towards these results was a systematic categorical analysis of the notions of nfold and infinite loop spaces. The results of this analysis will 449 be presented here. These include certain adjoint functor relationships that provide the conceptual reason that H.(~nsnx;zp) and H,(QX;Zp) are functors of H,(X;Zp) and that precisely relate maps between spaces to maps between spectra. These categorical considerations motivate the introduction of certain nonstandard categories, I and i, of (bounded) spectra and ~spectra, and the main purpose of this paper is to propagandize these categories. It is clear from their definitions that these categories are considerably easier to work with topologically than are the usual ones, but it is not clear that they are sufficiently large to be of interest. We shall remedy this by showing that, in a sense to be made precise, these categories are equivalent for the purposes of homotopy theory to the standard categories of (bounded) spectra and ~spectra. We extend the theory to unbounded spectra in the last section.
Effective generalized SeifertVan Kampen: how to calculate ΩX, preprint available at qalg/9710011
"... A central concept in algebraic topology since the 1970’s has been that of delooping machine [4] [23] [29]. Such a “machine ” corresponds to a notion of Hspace, or space with a multiplication satisfying associativity, unity and inverse properties up to homotopy in an appropriate way, including highe ..."
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Cited by 4 (1 self)
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A central concept in algebraic topology since the 1970’s has been that of delooping machine [4] [23] [29]. Such a “machine ” corresponds to a notion of Hspace, or space with a multiplication satisfying associativity, unity and inverse properties up to homotopy in an appropriate way, including higher order coherences as first investigated in [33]. A delooping machine is a specification of the extra homotopical structure carried by the loop space ΩX of a connected basepointed topological space X, exactly the structure allowing recovery of X by a “classifying space ” construction. The first level of structure is that the component set π0(ΩX) has a structure of group π1(X, x). Classically the SeifertVan Kampen theorem states that a pushout diagram of connected spaces gives rise to a pushout diagram of groups π1. The loop space construction ΩX with its delooping structure being the higherorder “topologized ” generalization of π1, an obvious question is whether a similar SeifertVan Kampen statement holds for ΩX. The aim of this paper is to describe the operation underlying pushout of spaces with loop space structure, answering the above question by giving a SeifertVan Kampen statement for delooping machinery. We work with Segal’s machine [28] [36]. Our SeifertVan
THE HOMOLOGY OF THE LITTLE DISKS OPERAD
, 2006
"... This expository paper aims to be a gentle introduction to the topology of configuration spaces, or equivalently spaces of little disks. The pantheon of topological spaces which beginning graduate students see is limited – spheres, projective spaces, products of such, perhaps some spaces such as Lie ..."
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Cited by 3 (1 self)
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This expository paper aims to be a gentle introduction to the topology of configuration spaces, or equivalently spaces of little disks. The pantheon of topological spaces which beginning graduate students see is limited – spheres, projective spaces, products of such, perhaps some spaces such as Lie groups, Grassmannians or knot complements. We would like for Euclidean configuration spaces to be added to
The Incommunicability of Content
 Mind
, 1966
"... 1. Setting up the foundations 3 2. The EilenbergSteenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6 ..."
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Cited by 2 (0 self)
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1. Setting up the foundations 3 2. The EilenbergSteenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6