You are here
Home > Topology

By Christoph Schweigert

Similar topology books

Riemannian Geometry: A Modern Introduction by Isaac Chavel PDF

Requiring in basic terms an figuring out of differentiable manifolds, Isaac Chavel covers introductory principles by means of a variety of extra really expert issues during this moment version. He offers a clearer remedy of many issues, with new proofs of a few theorems and a brand new bankruptcy at the Riemannian geometry of surfaces.

Read e-book online Elliptic Cohomology PDF

Elliptic cohomology is an exceptionally appealing thought with either geometric and mathematics points. the previous is defined by way of the truth that the speculation is a quotient of orientated cobordism localised clear of 2, the latter through the truth that the coefficients coincide with a hoop of modular kinds. the purpose of the e-book is to build this cohomology concept, and review it on classifying areas BG of finite teams G.

Read e-book online Local Cohomology: A seminar given by A. Grothendieck Harvard PDF

Shipped from united kingdom, please let 10 to 21 enterprise days for arrival. Lecture Notes in arithmetic forty-one. 106pp. good shape ex. lib.

Extra info for Algebraic Topology [Lecture notes]

Example text

Then W 2 /W 1 ∼ = 6 2 i=1 S . 12 Cellular homology In the following, X will always be a CW complex. 1. For the relative homology of the skeleta, we have Hq (X n , X n−1 ) = 0 for all q = n 1. Proof. Using the identification of relative homology and reduced homology of the quotient gives ˜ q (X n /X n−1 ) ∼ Hq (X n , X n−1 ) ∼ =H = ˜ q (Sn ). 8. 7. 2. Consider the inclusion in : X n → X of the n-skeleton X n into X. 1. The induced map Hn (in ) : Hn (X n ) → Hn (X) is surjective. 2. On the (n + 1)-skeleton we get an isomorphism Hn (in+1 ) : Hn (X n+1 ) ∼ = Hn (X).

The cone over an n-simplex is an (n+1)-simplex. The cone over Sn is a closed (n+1)-ball. 2. Note that for any topological space X, the cone CX is contractible to its apex. Thus ˜ n (CX, CA) = 0 ˜ n (CX) = 0 for all n 0. Similarly, for A ⊂ X, we have CA ⊂ CX and H H for all n 0. 3. The suspension of Sn is ΣSn ∼ = Sn+1 . 4. We have natural embeddings X → CX and CX → ΣX. We can see the suspension as two cones, glued together at their bases. 11 (Suspension isomorphism). Let A ⊂ X be a closed subspace and assume that A is a deformation retract of an open neighbourhood A ⊂ U .

Similarly, for A ⊂ X, we have CA ⊂ CX and H H for all n 0. 3. The suspension of Sn is ΣSn ∼ = Sn+1 . 4. We have natural embeddings X → CX and CX → ΣX. We can see the suspension as two cones, glued together at their bases. 11 (Suspension isomorphism). Let A ⊂ X be a closed subspace and assume that A is a deformation retract of an open neighbourhood A ⊂ U . Then ˜ n−1 (X, A), Hn (ΣX, ΣA) ∼ =H for all n > 0. Proof. 1. We first note two equivalences: X ∪ CA/CA A ⊂ CA: X/A, where the cone CA is attached to X by identifying A ⊂ X and ✁✄❈❆ ✁✄ ❈❆ ✁✄ ❈❆ CA ✁ ✄ ❈ ❆ ✁ ✄ ❈ ❆ ✁ ✄ ❈ ❆ A and CX/(CA ∪ X) ΣX/ΣA: r r r✁❆❅ r r✁ r ❆❅ r r r r✁ r rr❆r ❅ CX r✁r rr rr rr ❆rr ❅ r✁r rr rr rr rr r❆r r ❅ ✁r r r r r r r❆ ❅ X r r r✁❆❅r r✁ r❆❅ r✁r rr ❆rr ❅ ΣX r✁r rr rr ❆rr ❅ r✁r rr rr rr ❆rr ❅ ✁r r r r r ❆r ❅ ❅ ❆ rr rr rr rr r✁r ❅ ❆ rr rr rr r✁r ❅ ❆ rr rr r✁r ❅ ❆ r ✁r ❅❆r r✁r ❅ ❆✁ r CA ΣA 40 2.