# A Concise Course in Algebraic Topology by J. P. May

By J. P. May

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This is often an creation to geometric algebra, a substitute for conventional vector algebra that expands on it in ways:

1. as well as scalars and vectors, it defines new gadgets representing subspaces of any dimension.

2. It defines a product that is strongly stimulated by means of geometry and will be taken among any gadgets. for instance, the fabricated from vectors taken in a undeniable approach represents their universal plane.

This process was once invented by way of William Clifford and is frequently often called Clifford algebra. it is truly older than the vector algebra that we use at the present time (due to Gibbs) and comprises it as a subset. through the years, a number of components of Clifford algebra were reinvented independently via many of us who came upon they wanted it, frequently no longer knowing that every one these components belonged in a single procedure. this implies that Clifford had definitely the right inspiration, and that geometric algebra, no longer the diminished model we use at the present time, merits to be the traditional "vector algebra. " My target in those notes is to explain geometric algebra from that viewpoint and illustrate its usefulness. The notes are paintings in growth; i will continue including new subject matters as I examine them myself.

https://arxiv. org/abs/1205. 5935

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A criterion for a map to be a cofibration We want a criterion that allows us to recognize cofibrations when we see them. We shall often consider pairs (X, A) consisting of a space X and a subspace A. Cofibration pairs will be those pairs that “behave homologically” just like the associated quotient spaces X/A. Definition. A pair (X, A) is an NDR-pair (= neighborhood deformation retract pair) if there is a map u : X −→ I such that u−1 (0) = A and a homotopy h : X × I −→ X such that h0 = id, h(a, t) = a for a ∈ A and t ∈ I, and h(x, 1) ∈ A if u(x) < 1; (X, A) is a DR-pair if u(x) < 1 for all x ∈ X, in which case A is a deformation retract of X.

The starting point of the construction of general covers is the following description of regular covers and in particular of the universal cover. Proposition. Let p : E −→ B be a cover such that Aut(E) acts transitively on Fb . Then the cover p is regular and E/ Aut(E) is homeomorphic to B. Proof. For any points e, e′ ∈ Fb , there exists g ∈ Aut(E) such that g(e) = e′ and thus p∗ (π1 (E, e)) = p∗ (π1 (E, e′ )). Therefore all conjugates of p∗ (π1 (E, e)) are equal to p∗ (π1 (E, e)) and p∗ (π1 (E, e)) is a normal subgroup of π1 (B, b).

COVERINGS OF GROUPOIDS 23 (ii) Let C be a small groupoid. Define the star of x, denoted St(x) or StC (x), to be the set of objects of x\C , that is, the set of morphisms of C with source x. Write C (x, x) = π(C , x) for the group of automorphisms of the object x. (iii) Let E and B be small connected groupoids. A covering p : E −→ B is a functor that is surjective on objects and restricts to a bijection p : St(e) −→ St(p(e)) for each object e of E . For an object b of B, let Fb denote the set of objects of E such that p(e) = b.