In this largely expository note we give some homological properties of algebraic maps of complex algebraic varieties which are rather surprising from the topological point of view. These include a generalisation to higher dimension of the invariant cycle theorem for maps to curves. These properties are all corollaries of a recent deep theorem of Deligne, Gabber, Beilinson, and Bernstein which i...

This is a short note intended to explore the applications of duality theory to the study of manifolds. I discuss Alexander duality, Lefschetz duality and Poincare duality, along with applications to the study of compact connected orientable manifolds. 1. Manifolds and points 1.1. The core question. One of the questions we shall be interested in is: Given two manifolds M and N , what are the way...

Perspective (Invariants). The homotopy type of a fixed configuration space is a homeomorphism invariant of the background space, and, in the case of a manifold, it turns out that these invariants remember a rather large amount of information. A simple-minded example is provided by Euclidean spaces of different dimension; indeed, as we will see, there is a homotopy equivalence B2(R) ' B2(R) if a...

Nowadays, Internet appears as a suitable tool for performing scientific computations in a distributed collaborative way. One of the fields where this idea can be applied is that of Symbolic Computation. Computer Algebra packages are being extended to interconnect them. In fact, some Computer Algebra systems are already capable of performing distributed computing. An example is Distributed Maple...

Geometric β ⊆ α T (β) ≤ S(α) V vertices of K dimS = card(α)− 1 dimS = d dimA = maxα∈A(dimα) dimA = maxS∈A dimS Table 1. Analogous Properties of Abstract and Geometric Simplicial Complexes

Many propositions in spatial and qualitative reasoning can be modelled as regions in phase or con guration spaces Deciding the consistency of k propositions then amounts to deciding the question of whether there exists a point which simultaneously falls into all k corre sponding regions A more di cult problem is to decide whether there is a point which falls only within the given k regions I ca...

A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes the non-simply connected case, and...

This is a course in algebraic topology for anyone who has seen the fundamental group and homology. Traditionally (20 years ago), the syllabus included cohomology, the universal coefficient theorem, products, Tor and Ext and duality. We will cover all those topics, while also building a modern framework of derived categories and derived functors using the Cartan–Eilenberg method. We will apply t...