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158
Deriving Dg Categories
, 1993
"... We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categori ..."
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Cited by 123 (9 self)
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We investigate the (unbounded) derived category of a differential Zgraded category (=DG category). As a first application, we deduce a 'triangulated analogue` (4.3) of a theorem of Freyd's [5, Ex. 5.3 H] and Gabriel's [6, Ch. V] characterizing module categories among abelian categories. After adapting some homological algebra we go on to prove a 'Morita theorem` (8.2) generalizing results of [19] and [20]. Finally, we develop a formalism for Koszul duality [1] in the context of DG augmented categories. Summary We give an account of the contents of this paper for the special case of DG algebras. Let k be a commutative ring and A a DG (k)algebra, i.e. a Zgraded kalgebra A = a p2Z A p endowed with a differential d of degree 1 such that d(ab) = (da)b + (\Gamma1) p a(db) for all a 2 A p , b 2 A. A DG (right) Amodule is a Zgraded Amodule M = ` p2Z M p endowed with a differential d of degree 1 such that d(ma) = (dm)a + (\Gamma1) p m(da) for all m 2 M p , a 2 A. A morphism of DG Amodules is a homogeneous morphism of degree 0 of the underlying graded Amodules commuting with the differentials. The DG Amodules form an abelian category CA. A morphism f : M ! N of CA is nullhomotopic if f = dr + rd for some homogeneous morphism r : M ! N of degree1 of the underlying graded Amodules.
Ganea's conjecture on LusternikSchnirelmann category
, 1997
"... A series of complexes Qp indexed by all primes p is constructed with catQp = 2 and catQpSn = 2 for either n 2 or n = 1 and p = 2. This disproves Ganea’s conjecture ..."
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Cited by 27 (7 self)
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A series of complexes Qp indexed by all primes p is constructed with catQp = 2 and catQpSn = 2 for either n 2 or n = 1 and p = 2. This disproves Ganea’s conjecture
Relations amongst Toda brackets and the Kervaire invariant in dimension 62
 J. London Math. Soc
, 1984
"... In this paper we use relations amongst Toda brackets and a lot of detailed information about the homotopy groups of spheres to show that there exists a 62dimensional framed manifold with Kervaire invariant one. This paper, together with [4, 5], represents an effort to supply full details for a numb ..."
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Cited by 26 (2 self)
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In this paper we use relations amongst Toda brackets and a lot of detailed information about the homotopy groups of spheres to show that there exists a 62dimensional framed manifold with Kervaire invariant one. This paper, together with [4, 5], represents an effort to supply full details for a number of the results
Embedding and knotting of manifolds in Euclidean spaces
 London Math. Soc. Lect. Notes
"... Abstract. A clear understanding of topology of higherdimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higherdimensional topology in a way which makes clear the visual and algebraic constructions appear natu ..."
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Cited by 24 (9 self)
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Abstract. A clear understanding of topology of higherdimensional objects is important in many branches of both pure and applied mathematics. In this survey we attempt to present some results of higherdimensional topology in a way which makes clear the visual and algebraic constructions appear naturally in the study of geometric problems. Before giving a general construction, we illustrate the main ideas in simple but important particular cases, in which the essence is not veiled by technicalities. More specifically, we present several classical and modern results on the embedding and knotting of manifolds in Euclidean space. We state many concrete results (in particular, recent explicit classification of knotted tori). Their statements (but not proofs!) are simple and accessible to nonspecialists. We outline a general approach to embeddings via the classical van KampenShapiroWuHaefligerWeber ’deleted product ’ obstruction. This approach reduces the isotopy classification of embeddings to the homotopy classification of equivariant maps, and so implies the above concrete results. We describe the revival of interest in this beautiful branch of topology, by presenting new results in this area (of Freedman, Krushkal, Teichner, Segal, Spie˙z and the author): a generalization the HaefligerWeber embedding theorem below the metastable dimension range and examples showing that other analogues of this theorem are false outside the metastable dimension range. 1.
Nielsen coincidence theory in arbitrary codimensions
 J. reine angew. Math
"... Let f1,f2: M − → N be two (continuous) maps between smooth connected manifolds M and N without boundary, of strictly positive dimensions m and n, resp., M being compact. We are interested in making the coincidence locus C(f1,f2): = {x ∈ M  f1(x) = f2(x)} as small (or simple in some sense) as possi ..."
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Cited by 22 (5 self)
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Let f1,f2: M − → N be two (continuous) maps between smooth connected manifolds M and N without boundary, of strictly positive dimensions m and n, resp., M being compact. We are interested in making the coincidence locus C(f1,f2): = {x ∈ M  f1(x) = f2(x)} as small (or simple in some sense) as possible after possibly deforming f1 and f2 by a homotopy. Question. How large is the minimum number of coincidence components MCC(f1,f2): = min{#π0(C(f ′ 1,f ′ 2))  f ′ 1 ∼ f1,f ′ 2 ∼ f2}? In particular, when does this number vanish, i.e. when can f1 and f2 be deformed away from one another? This is a very natural generalization of one of the central problems of classical fixed point theory (where M = N and f2 = identity map): determine the minimum number of fixed points among all maps in a given homotopy class (see [Br] and [BGZ], proposition 1.5). Note, however, that in higher codimensions m − n> 0 the coincidence locus is generically a closed (m−n)manifold so that it makes more sense to count pathcomponents rather than points. Also the methods of (first order, singular) (co)homology will no longer be strong enough to capture the subtle geometry of coincidence manifolds. In this lecture I will use the language of normal bordism theory (and a nonstabilized version thereof) to define and study lower bounds N(f1,f2) (and N #(f1,f2)) for MCC(f1,f2). After performing an approximation we may assume that the map (f1,f2) : M → N × N is smooth and transverse to the diagonal ∆ = {(y,y) ∈ N × N  y ∈ N}. Then the coincidence locus C = C(f1,f2) = (f1,f2) −1 (∆) is a closed smooth (m − n)dimensional manifold, equipped with i) maps
Homotopy theory of the suspensions of the projective plane
 Memoirs AMS
"... Abstract. The homotopy theory of the suspensions of the real projective plane is largely investigated. The homotopy groups are computed up to certain range. The decompositions of the self smashes and the loop spaces are studied with some applications to the Stiefel manifolds. ..."
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Cited by 20 (11 self)
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Abstract. The homotopy theory of the suspensions of the real projective plane is largely investigated. The homotopy groups are computed up to certain range. The decompositions of the self smashes and the loop spaces are studied with some applications to the Stiefel manifolds.
A∞method in LusternikSchnirelmann category
, 2004
"... To clarify the method behind [11], a generalisation of BersteinHilton Hopf invariants is defined as ‘higher Hopf invariants’. They detect the higher homotopy associativity of Hopf spaces and are studied as obstructions not to increase the LS category by one by attaching a cone. Under a condition be ..."
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Cited by 14 (9 self)
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To clarify the method behind [11], a generalisation of BersteinHilton Hopf invariants is defined as ‘higher Hopf invariants’. They detect the higher homotopy associativity of Hopf spaces and are studied as obstructions not to increase the LS category by one by attaching a cone. Under a condition between dimension and LS category, a criterion for Ganea’s conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of [11] for all the cases except the case when p = 2. As an application, conditions in terms of homotopy invariants of the characteristic maps
A braided simplicial group
 Proc. London Math. Soc
"... Abstract. By studying braid group actions on Milnor’s construction of the 1sphere, we show that the general higher homotopy group of the 3sphere is the fixed set of the pure braid group action on certain combinatorially described group. We also give certain representation of higher differentials i ..."
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Cited by 13 (10 self)
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Abstract. By studying braid group actions on Milnor’s construction of the 1sphere, we show that the general higher homotopy group of the 3sphere is the fixed set of the pure braid group action on certain combinatorially described group. We also give certain representation of higher differentials in the Adams spectral sequence for π∗(S 2). 1.