With firm foundations dating only from the 1950s, algebraic topology is a relatively young area of mathematics. There are very few textbooks that treat fundamental topics beyond a first course, and many topics now essential to the field are not treated in any textbook. J. Peter May's "A Concise Course in Algebraic Topology" addresses the standard first course material, such as fundamental groups, covering spaces, the basics of homotopy theory, and homology and cohomology. In this sequel, May and his coauthor, Kathleen Ponto, cover topics that are essential for algebraic topologists and others interested in algebraic topology, but that are not treated in standard texts. They focus on the localization and completion of topological spaces, model categories, and Hopf algebras. The first half of the book sets out the basic theory of localization and completion of nilpotent spaces, using the most elementary treatment the authors know of. It makes no use of simplicial techniques or model categories, and it provides full details of other necessary preliminaries. With these topics as motivation, most of the second half of the book sets out the theory of model categories, which is the central organizing framework for homotopical algebra in general. Examples from topology and homological algebra are treated in parallel. A short last part develops the basic theory of bialgebras and Hopf algebras.

More Concise Algebraic Topology: Localization, Completion, and Model Categories

In the renaming task n+1 processes start with unique input names from a large space and must choose unique output names taken from a smaller name space, namely 0,1,...,K. To rule out trivial solutions, a protocol must be anonymous: the value chosen by a process can depend on its input name and on the execution, but not on the specific process id. Attiya et al. showed in 1990 that renaming has a wait-free solution when K

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New combinatorial topology upper and lower bounds for renaming

The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.

Modern Classical Homotopy Theory

Preface Part I. Cubical Diagrams: 1. Preliminaries 2. 1-cubes: homotopy fibers and cofibers 3. 2-cubes: homotopy pullbacks and pushouts 4. 2-cubes: the Blakers-Massey Theorems 5. n-cubes: generalized homotopy pullbacks and pushouts 6. The Blakers-Massey Theorems for n-cubes Part II. Generalizations, Related Topics, and Applications: 7. Some category theory 8. Homotopy limits and colimits of diagrams of spaces 9. Cosimplicial spaces 10. Applications Appendix References Index.

Cubical Homotopy Theory

This paper is devoted to analyze the class of initial data that can be insensitized for the heat equation.
 This issue has been extensively addressed in the literature
 both
 in the case of complete and approximate insensitization (see [19] and [1], respectively).

 But in the context of pure insensitization there are very few results identifying the class of initial data that
 can be insensitized. This is a delicate issue which is related to the fact that
insensitization turns out to be equivalent to suitable
observability estimates for a coupled system of heat equations,
one being forward and the other one backward in time.
 The existing Carleman inequalities techniques can be applied but they only give interior information of the solutions,
 which hardly allows identifying the initial data because of the strong irreversibility of the equations involved in the
 system, one of them being an obstruction at the initial time $t=0$ and the other one at the final one $t=T$.

 

In this article we consider different geometric configurations in which the subdomains to be insensitized and the one in which the external control acts play a key role. We show that, under rather restrictive geometric restrictions, initial data in a
class that can be characterized in terms of a summability condition of their Fourier coefficients with suitable weights, can be
 insensitized. But, the main result of the paper, which might seem surprising, shows that this fails to be true in general, so that even the first eigenfunction of the system can not be
 insensitized. This result is similar to those obtained in the context of the null controllability of the heat equation in unbounded domains in [14] where it is shown that smooth and compactly supported initial data may not be controlled.

 

Our proofs combine the existing observability results for heat equations obtained by means of Carleman inequalities, energy and gaussian estimates and Fourier expansions.

Identification of the class of initial data for the insensitizing control of the heat equation

Introduction.- Basic Concepts and Notation.- Function Spaces.- Connectedness and Algebraic Invariants.- Homotopy Groups.- Homotopy Extension and Lifting Properties.- CW-Complexes Homology.- Homotopy Properties of CW-Complexes.- Cohomology Groups and Related Topics.- Vector Bundles.- K-Theory.- Adams Operations and Applications.- Relations Between Cohomology and Vector Bundles.- Cohomology Theories and Brown Representability.- Appendix A: Proof of the Dold-Thom Theorem.- Appendix B: Proof of the Bott Periodicity Theorem.- References.- Index.- Glossary.

Algebraic Topology from a Homotopical Viewpoint

Equivariant homotopical homology with coefficients in a Mackey functor

Making use of a modified version, due to McCord, of the Dold-Thom construction of ordinary homology, we give a simple topological definition of a transfer for ramified covering maps in homology with arbitrary coefficients. The transfer is induced by a suitable map between topological groups. We also define a new cohomology transfer which is dual to the homology transfer. This duality allows us to show that our homology transfer coincides with the one given by L. Smith. With our definition of the homology transfer we can give simpler proofs of the properties of the known transfer and of some new ones. Our transfers can also be defined in Karoubi's approach to homology and cohomology. Furthermore, we show that one can define mixed transfers from other homology or cohomology theories to the ordinary ones.

Marcelo A. Aguilar

Papers

Algebraic Topology from a Homotopical Viewpoint

Equivariant homotopical homology with coefficients in a Mackey functor

Transfers for ramified covering maps in homology and cohomology

A classification of cohomology transfers for ramified covering maps

Characteristic classes and transversality