For the beautiful world !

Welcome to my blog! Contact me via "quiddite[number four]dino[at]gmail[dot]com"

2025-2026-1

2025/09/12

This md. file is devoted to gather (most of) the materials for the courses I take in this semester, both for preparation and review.

QC-syl

  week 1-12

MM-syl

RS-syl

lecture slides

TA-syl

  topics

KI-syl

lecture slides

More

MSRS GT3M

G2T learning materials

2025/06/07

Some frequently used constructions (functors) and tools in AT

2025/05/25

spaces

§ concerning groups
Moore space M(G, n)
    • definition M(G, n):=the CW complex with $H_i(-)=\left.\begin{cases} 0&,i\neq n\\ G&,i=n \end{cases}\right.$, we may also assume π1(−) = 0 if n > 0.
    • basic results
    • applications
EM space K(G, n)
    • definition K(G, n):=the CW complex with $\pi_i(-)=\left.\begin{cases} 0&,i\neq n\\ G&,i=n \end{cases}\right.$.
    • basic results
      • unique up to (weak) homotopy equivalence;
      • product: K(G, n) × K(H, n) ≃ K(G × H, n)
      • representation results: Let G be an Ab-grp, X a CW complex, then ⟨X, K(G, n)⟩ ↔︎ Hn(X; G), [f] ↦ f*α where α ∈ Hn(K; G)) = Hom(Hn(K), G) is given by the inverse of Hurewitz isomorphism G → Hn(K).
      • cohomology: $H^*(K(\mathbb{Z},n);\mathbb{Q})=\left.\begin{cases} \mathbb{Q}[x_n]&,n\text{ is even}\\ \mathbb{Q}[x_n]/(x_n^2)&,n\text{ is odd} \end{cases}\right.$, using SSS with loopspace fibration, see page 11 of AT-all.
    • applications
      • motivations of (complex) universal bundles: K(ℤ, 1) = S1, K(ℤ/2, 1) = ℝℙ∞, K(ℤ, 2) = ℂℙ∞.
      • (co)homology of groups: Hn(G) = Hn(K(G, n))
classifying space BG
    • definition
      • BG is the CW complex such that there is a principle G-bundle EG → BG, with EG weakly contractible
      • construction (Milnor): for a topo-grp G, there is a accenting chain via topological join: G ⊂ G⋆2 ⊂ ⋯ ⊂ G⋆n⋯ Let EG = ∪G⋆n, BG = EG/G.
    • basic results
      • uniqueness up to homotopy equivalence
      • bijection: $[B,BG]\leftrightarrow Bun_{G}(B)=\left.\begin{cases}\text{principal } G-\\\text{bundles over B}\end{cases}\right\} /\cong$
    • applications
Postnikov Towers
§ (direct) constructions
connected sum ⋅#⋅
topological join ⋅ ⋆ ⋅

    • definition M ⋆ N = M × [0, 1] × N/∼, where (x, 0, y1) ∼ (x, 0, y2), (x1, 1, y) ∼ (x2, 1, y)

    • basic results

      • if M is m-connented, N is n-connected, then M ⋆ N is (m + n + 2)-connected.
      • S0 ⋆ S0 = S1, Sm ⋆ Sn = Sm + n + 1.
      • M⋆n is always (n − 2)-connected.

    • applications

      • Milnor’s construction of classifying space: EG = ∪G⋆n, BG = EG/G
suspension ΣX
wedge sum ⋅ ∨ ⋅
smash product ⋅ ∧ ⋅
path space PX
loop space ΩX
Thom space TE
Configuration space ConfnX
Čech nerve
§ concerning mappings
mapping cylinder
mapping cone Cf
mapping torus
lens space S−/ℤ•
Hopf torus

tools

  • fiber bundle
  • Gysin sequence
  • Leray-Hirsch
  • Thom class
  • Steenrod square

2024-2025-2

2025/05/01

Useful links

2025/03/10
Homepages
Webpages
Lifestyles & Misc

QE

2025/03/01

Resources

some repositories: link
A&N

Harvard, UCLA, CUNY, Stanford, Maryland,

Florida, Rochester, Georgia, Hopkins, Northwestern, Rice

P&S(prob le m s) ⋄ Probability theory ⋄ Statistics
yau contest(10-13)
G&T(pro ble m s )

UCLA, Harvard, CUNY, Northwestern

partial solution to UCLA , the written qual book

⋄ Riemannian geometry (Page) ⋄ Algebraic topology ⋄ Misc

linear algebra

2024/12/08
The following file comes from 4 tutorial sessions I gave during the 2023-2024 academic year. 理科高等代数串讲

2024-2025-1

2024/12/03

This md. file is devoted to gather (most of) the materials for the courses I took in this semester.

RG

self-made notes RG
lecture notes
homeworks
my answers
solutions
books

QM

books
lecture slides
homeworks
my answers
solutions

AG

self-made notes: still on its way… AG
books
homeworks
materials
my answers
solutions

TT

books
final project

SG

lecture notes
books

Hello World

2024/11/17

Welcome to Hexo! This is my very first post. Check documentation for more info. If you get any problems when using Hexo, you can find the answer in troubleshooting or you can ask me on GitHub.

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Nevermore.