Download A Concise Course in Algebraic Topology by J. P. May PDF

By J. P. May

Algebraic topology is a easy a part of sleek arithmetic, and a few wisdom of this zone is imperative for any complex paintings when it comes to geometry, together with topology itself, differential geometry, algebraic geometry, and Lie teams. This e-book offers a close remedy of algebraic topology either for academics of the topic and for complicated graduate scholars in arithmetic both focusing on this region or carrying on with directly to different fields. J. Peter May's process displays the big inner advancements inside algebraic topology during the last numerous many years, so much of that are principally unknown to mathematicians in different fields. yet he additionally keeps the classical shows of assorted themes the place applicable. such a lot chapters finish with difficulties that extra discover and refine the ideas offered. the ultimate 4 chapters offer sketches of considerable parts of algebraic topology which are in general passed over from introductory texts, and the ebook concludes with an inventory of steered readings for these drawn to delving additional into the field.

Show description

Read or Download A Concise Course in Algebraic Topology PDF

Similar algebraic geometry books

Solitons and geometry

During this e-book, Professor Novikov describes fresh advancements in soliton idea and their kin to so-called Poisson geometry. This formalism, that's concerning symplectic geometry, is intensely valuable for the research of integrable structures which are defined when it comes to differential equations (ordinary or partial) and quantum box theories.

Algebraic Geometry Iv Linear Algebraic Groups Invariant Theory

Contributions on heavily comparable topics: the speculation of linear algebraic teams and invariant conception, through recognized specialists within the fields. The ebook can be very worthy as a reference and examine advisor to graduate scholars and researchers in arithmetic and theoretical physics.

Vector fields on singular varieties

Vector fields on manifolds play an immense position in arithmetic and different sciences. particularly, the Poincaré-Hopf index theorem offers upward thrust to the idea of Chern sessions, key manifold-invariants in geometry and topology. it's average to invite what's the ‘good’ suggestion of the index of a vector box, and of Chern periods, if the underlying house turns into singular.

Algebraic Topology

E-book by way of

Extra resources for A Concise Course in Algebraic Topology

Example text

38 GRAPHS Proof. We may write G = F/N for some free group F and normal subgroup N . As above, we may realize the inclusion of N in F by passage to fundamental groups from a cover p : E −→ B. Define the (unreduced) cone on E to be CE = (E × I)/(E × {1}) and define X = B ∪p CE/(∼), where (e, 0) ∼ p(e). Let U and V be the images in X of B ∐ (E × [0, 3/4)) and E × (1/4, 1], respectively, and choose a basepoint in E × {1/2}. Since U and U ∩ V are homotopy equivalent to B and E via evident deformations and V is contractible, a consequence of the van Kampen theorem gives the conclusion.

Choose x ∈ X, let b = f (x), and choose e ∈ Fb . There exists a map g : X −→ E such that g(x) = e and p ◦ g = f if and only if f∗ (π1 (X, x)) ⊂ p∗ (π1 (E, e)) in π1 (B, b). When this condition holds, there is a unique such map g. Proof. If g exists, its properties directly imply that im(f∗ ) ⊂ im(p∗ ). Thus assume that im(f∗ ) ⊂ im(p∗ ). Applied to the covering Π(p) : Π(E) −→ Π(B), the analogue for groupoids gives a functor Π(X) −→ Π(E) that restricts on objects to the unique map g : X −→ E of sets such that g(x) = e and p ◦ g = f .

Define E to be the set of equivalence classes of paths f in B that start at b and define p : E −→ B by p[f ] = f (1). Of course, the equivalence relation is homotopy through paths from b to a given endpoint, so that p is well defined. Thus, as a set, E is just StΠ(B) (b), exactly as in the construction of the universal cover of Π(B). The topology of B has a basis consisting of path connected open subsets U such that π1 (U, u) −→ π1 (B, u) is trivial for all u ∈ U . Since every loop in U is equivalent in B to the trivial loop, any two paths u −→ u′ in such a U are equivalent in B.

Download PDF sample

Rated 4.96 of 5 – based on 29 votes