I will discuss recent progress by many people in the program of extending natural topological invariants from manifolds to singular spaces. Intersection homology theory and mixed Hodge theory are model examples of such invariants. The past 20 years have seen a series of new invariants, partly inspired by string theory, such as motivic integration and the elliptic genus of a singular variety. Th...

1. Setting up the foundations 3 2. The Eilenberg-Steenrod axioms 4 3. Stable and unstable homotopy groups 5 4. Spectral sequences and calculations in homology and homotopy 6 5. Steenrod operations, K(π, n)’s, and characteristic classes 8 6. The introduction of cobordism 10 7. The route from cobordism towards K-theory 12 8. Bott periodicity and K-theory 14 9. The Adams spectral sequence and Hopf...

1. Given a map f :X→Y , show that there exists a map g :Y→X with gf ≃ 1 iff X is a retract of the mapping cylinder Mf . 2. (a) Suppose a CW complex X is the union of a finite number of subcomplexes Xi and that a subcomplex A of X is the union of subcomplexes Ai ⊂ Xi . Show that if each Xi deformation retracts onto Ai and each intersection of a subcollection of the Xi ’s deformation retracts ont...

1. Simplices, ∆-Complexes, and Homology: 1/8/15 1 2. Properties of Singular Homology: 1/13/15 4 3. Homotopy Invariance of Singular Homology: 1/15/15 7 4. Applications of Homotopy Invariance and Excision: 1/20/15 10 5. Equivalence of Singular and Simplicial Homology: 1/22/15 14 6. Degrees of Maps on Sn: 1/27/15 17 7. The Mayer-Vietoris Sequence and Applications: 1/29/15 20 8. CW Complexes: 2/3/1...

In this paper, we present a new version of cochains in Algebraic Topology, starting with “quantum differential forms”. This version provides many examples of modules over the braid group, together with a control of the non commutativity of cup-products on the cochain level. If the quantum parameter q is equal to 1, we essentially recover the commutative differential graded algebra of de Rham-Su...

These are the lecture notes for an Honours course in algebraic topology. They are based on standard texts, primarily Munkres’s “Elements of algebraic topology” and to a lesser extent, Spanier’s “Algebraic topology”. 1 What’s algebraic topology about? Aim lecture: We preview this course motivating it historically. Recall that a continuous map f : X −→ Y of topological spaces is a homeomorphism i...

The appearance of the complete Heyting algebra in the realm of Algebraic Topology is the main topic of the paper.

This short course on Algebraic Topology shall give an introduction to some essential concepts of this eld of mathematics: the fundamental group and homology theories. As an example we will show how those results are used to prove that the Euclidean spaces Rn and Rm are not homeomorphic for n 6= m, which is not possible to prove in a purely topological way.