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Crossed Complexes And Homotopy Groupoids As Non Commutative Tools For Higher Dimensional LocalToGlobal Problems
"... We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections. ..."
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We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections.
TORSION INVARIANTS OF SpincSTRUCTURES ON 3MANIFOLDS
"... Recently there has been a surge of interest in the SeibergWitten invariants of 3manifolds, see [3], [4], [7]. The SeibergWitten invariant of a closed oriented 3manifold M is a function SW from the set of Spin cstructures on M to Z. This function is defined under the assumption b1(M) ≥ 1 where ..."
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Cited by 10 (2 self)
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Recently there has been a surge of interest in the SeibergWitten invariants of 3manifolds, see [3], [4], [7]. The SeibergWitten invariant of a closed oriented 3manifold M is a function SW from the set of Spin cstructures on M to Z. This function is defined under the assumption b1(M) ≥ 1 where b1(M) is the
On Yetter’s invariant and an extension of the DijkgraafWitten invariant to categorical groups
 Theory Appl. Categ
"... We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the ..."
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We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module G. We use this interpretation to define a twisting of Yetter’s Invariant by cohomology classes of crossed modules, defined
Concordance spaces, higher simple homotopy theory, and applications
 Proc. Sympos. Pure
, 1978
"... While much is now known, through surgery theory, about the classification problem for manifolds of dimension at least five, information about the automorphism groups of such manifolds is as yet rather sparse. In fact, it seems that there is not a single closed manifold M of dimension greater than th ..."
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While much is now known, through surgery theory, about the classification problem for manifolds of dimension at least five, information about the automorphism groups of such manifolds is as yet rather sparse. In fact, it seems that there is not a single closed manifold M of dimension greater than three for which the homotopy type of the automorphism space Diff(M), PL(M), or TOP(M) in the smooth, PL, or topological category, respectively, is in any sense known. (As usual, Diff(M) is given the C° ° topology, PL(M) is a simplicial group, and TOP(M) is the singular complex of the homeomorphism group with the compactopen topology.) Besides surgery theory, the principal tool in studying homotopy properties of these automorphism spaces is the concordance space functor C(M) = {automorphisms of M x /fixed on M x 0}. This paper is a survey of some of the main results to date on concordance spaces. Here is an outline of the contents. In §1 we describe how, in a certain stable dimension range, C{M) is a homotopy functor of M, which we denote by ^(M). The application to automorphism spaces is outlined in §2. In §3 we recall the explicit calculations which have been made for %$?(M) and %{g(M), along the lines pioneered by Cerf, and apply them in §4 to compute the group of isotopy classes of automorphisms of the «torus, n ^ 5. §5 is concerned with a stabilized version of #(Af), defined roughly as Q^iS^M), together with the curious equivalence of D^PLOS^M) with ^PL(M)/^Diff(M), due to BurgheleaLashof (based on earlier fundamental work of Morlet). In §6, ^PL(M) is "reduced " to higher simplehomotopy theory. This has some interest in its own right, e.g., it provides a fibered form of Wall's obstruction to finiteness. The important new work of Waldhausen relating <^PL(M) to algebraic ATtheory is outlined, very briefly and imperfectly, in §7. This seems to be the most promising area for future developments in the sub
Crossed complexes, and free crossed resolutions for amalgamated sums and HNNextensions of groups
 Georgian Math. J
, 1999
"... Dedicated to Hvedri Inassaridze for his 70th birthday The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the p ..."
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Cited by 7 (6 self)
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Dedicated to Hvedri Inassaridze for his 70th birthday The category of crossed complexes gives an algebraic model of CWcomplexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating nonabelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions for amalgamated sums and HNNextensions of groups, and so obtain computations of higher homotopical syzygies in these cases. 1
2004, ‘The local structure of algebraic Ktheory
"... Algebraic Ktheory draws its importance from its effective codification of a mathematical phenomenon which occurs in as separate parts of mathematics as number theory, geometric topology, operator algebra, homotopy theory and algebraic geometry. In reductionistic language the phenomenon can be phras ..."
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Algebraic Ktheory draws its importance from its effective codification of a mathematical phenomenon which occurs in as separate parts of mathematics as number theory, geometric topology, operator algebra, homotopy theory and algebraic geometry. In reductionistic language the phenomenon can be phrased as there is no canonical choice of coordinates. As such, it is a metatheme for mathematics, but the successful codification of this phenomenon in homotopytheoretic terms is what has made algebraic Ktheory into a valuable part of mathematics. For a further discussion of algebraic Ktheory we refer the reader to chapter I below. Calculations of algebraic Ktheory are very rare, and hard to get by. So any device that allows you to get new results is exciting. These notes describe one way to get such results. Assume for the moment that we know what algebraic Ktheory is, how does it vary with its input? The idea is that algebraic Ktheory is like an analytic function, and we have this other analytic function called topological cyclic homology (T C) invented by Bökstedt, Hsiang and
Simple Homotopy Types and Finite Spaces
, 2006
"... We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X ց Y of finite sp ..."
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Cited by 4 (2 self)
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We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X ց Y of finite spaces induces a simplicial collapse K(X) ց K(Y) of their associated simplicial complexes. Moreover, a simplicial collapse K ց L induces a collapse X(K) ց X(L) of the associated finite spaces. This establishes a onetoone correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. Furthermore, this class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space.
The Embedding of Homotopy Types into Manifolds
"... This paper was dated November, 1965; Princeton University. ..."
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This paper was dated November, 1965; Princeton University.
NORMALISATION FOR THE FUNDAMENTAL CROSSED COMPLEX OF A SIMPLICIAL SET
, 2007
"... Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. Th ..."
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Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes.
ONEPOINT REDUCTIONS OF FINITE SPACES, HREGULAR CWCOMPLEXES AND COLLAPSIBILITY
, 801
"... Abstract. We investigate onepoint reduction methods of finite topological spaces and their relationship with collapsibility of complexes. We also introduce the notion of hregular CWcomplex, generalizing the concept of regular CWcomplex and prove that the hregular CWcomplexes are modeled, up to ..."
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Abstract. We investigate onepoint reduction methods of finite topological spaces and their relationship with collapsibility of complexes. We also introduce the notion of hregular CWcomplex, generalizing the concept of regular CWcomplex and prove that the hregular CWcomplexes are modeled, up to homotopy, by their associated finite spaces (or face posets). This is accomplished by generalizing a classical result of McCord on simplicial complexes. 1.