The concept of fibration is one of the great unifying mathematical ideas. It was initially introduced around 1930 in geometry and topology, and gradually expanded into many other parts of mathematics. Together with fibre bundles (which precedeed fibrations), they give formal expression to the idea of a continuous family of spaces, and of operations on such families. This monograph contains an exposition of the fundamental ideas of the theory of fibrations with particular emphasis on their classification. It deals at length with various types of fibrations as defined by Hurewicz, Dold and Serre, as well as the quasifibrations of Dold and Thom. The relationship between these concepts is analyzed in depth, with examples and counter-examples given. One of the salient properties of fibre bundles is that they are classified by homotopy classes of maps into some special spaces called classifying spaces. The classifying theory for fibrations is presented both abstractly, through the theory of representable functors, and constructively, by describing various models, like those introduced by Dold and Lashof, and by Milgram and Steenrod. In the couple of decades following their intoduction, the growth of the theory of fibrations resulted in a plethora of similar and interrelated theories and classification results for vector bundles, general fibre bundles, and other types of fibre spaces. As a new organizational principle, Peter May invented the concept of F-fibrations that generalizes all of the above, and is at the same time sufficiently structured to admit workable classification objects. The second part of the book is dedicated to an in-depth discussion of the theory of F-fibrations. The book is reasonably self-contained and the reader is assumed to have only some knowledge of general topology and basic homotopy theory, including elementary properties of homotopy groups. However, one must be aware that the level of exposition is at some places more advanced, and for these a prior course in algebraic topology or in the theory of fibre bundles would be very helpful, both as a motivation for the problems that are studied, as well as a measure of the required mathematical sophistication. The book can be used both as a text-book or as a reference. Most chapters are concluded with historical notes, tracing the origins of the concepts and the developments related to the classification of fibre bundles and fibrations.
F.02 Acquisition of new scientific knowledge
COBISS.SI-ID: 16616793This was a series of plenary invited talks at the international meeting on geometric topology, organizedin Kobe, Japan, at which our postdoc researcher presented some of our main new results. Here are the title of his lectures: 1. Cayley graphs and basic group invariants 2. Asymptotic dimension and Švarc-Milnor Lemma 3. Hyperbolic groups 4. Asymptotic cones His talks were received by great interest and intensive discussions followed after every presentation. This visit was organized within the framework of our longterm collaboration with Japanese geometric topologists, led by Katsuya Eda at the Waseda University in Tokyo.
B.04 Guest lecture
COBISS.SI-ID: 16560473The relationship between \lor-decompositions of a coH-space C and decompositions of the identity as a sum of pairwise orthogonal idempotents in the image of the representation [C,C] \to \text{End}(H_\ast(G)) is studied. We show that the image of this representation is a semiperfect ring for every finite p-local coH-space C, which implies a strong form of Wilkerson's theorem. The discussion is based on the description of the algebraic structure [C,C], which is called a loop near-ring. Local loop near-rings are introduced, and a theorem of unique \lor-decompositions of coH-spaces, which is completely analogous to the Krull-Schidt-Azumaya theorem for modules, is proven. Dual results hold for H-spaces.
D.09 Tutoring for postgraduate students
COBISS.SI-ID: 16196441We present an algorithm which produces a decomposition of a regular finite cellular complex with a discrete Morse function analogous to the Morse-Smale decomposition of a smooth manifold with respect to a smooth Morse function. The advantage of our algorithm compared to similar existing results is that it works, at least theoretically, in any dimension. Practically, there are dimensional restrictions due to the size of cellular complexes of higher dimensions, though. We prove that the algorithm is correct in the sense that it always produces a decomposition into descending and ascending regions of the critical cells in a finite number of steps, and that, after a finite number of subdivisions, all the regions are topological discs. The efficiency of the algorithm is discussed and its performance on several examples is demonstrated. In the last chapter we turn our attention to infinite and locally finite complexes of finite dimension and discrete vector fields on them. We present sufficient conditions for a discrete vector field V on such a complex to arise from a proper discrete Morse function.
D.09 Tutoring for postgraduate students
COBISS.SI-ID: 15185241The notion of topological complexity was introduced by Farber in 2001 as a measure of discontinuity of motion planning algorithms on a configuration space of a robot. Some seven years later Iwase and Sakai presented the notion of monoidal topological complexity that they have shown to be a special case of the fibrewise pointed Lusternik-Schnirelmann category. Hence, numerous methods that work for the LS category can be reused in this fibrewise setting in order to obtain the analoguous results for the monoidal topological complexity. We derive an upper bound for the topological complexity of a mapping telescope almost directly from the fibrewise definition. With a bit more effort we find two alternative fibrewise definitions of monoidal topological complexity, which are analogous to the Whitehead and Ganea definitions of the LS category. From a fibrewise diagram connecting the two definitions we obtain several lower bounds, in particular the stable and the weak topological complexity. We compare these lower bounds to one another and consider some cases where equalities occur.
D.09 Tutoring for postgraduate students
COBISS.SI-ID: 263847936