Algebraic topology Study Guide

📖 Core Concepts Algebraic invariant – a group, ring, or module attached to a space that never changes under homeomorphism or homotopy equivalence. Homotopy equivalence – a weaker “same shape” relation: two spaces \(X,Y\) are homotopy equivalent if there are maps \(f:X\to Y,\;g:Y\to X\) with \(g\circ f\simeq \text{id}X\) and \(f\circ g\simeq \text{id}Y\). Fundamental group \(\pi1(X)\) – the first homotopy group; records how loops can be deformed in \(X\). Non‑abelian in general. Homology groups \(Hn(X)\) – abelian groups measuring \(n\)-dimensional “holes”. The free rank of \(Hn\) is the \(n\)‑th Betti number \(bn\). Cohomology groups \(H^n(X)\) – obtained by dualizing the chain complex; carry extra structure (cup product) and often easier to compute functorially. CW complex – a space built from cells glued inductively; the preferred setting for calculations because it needs few cells and behaves well categorically. Functoriality – a continuous map \(f:X\to Y\) induces homomorphisms \(f:Hn(X)\to Hn(Y)\) and \(f^:H^n(Y)\to H^n(X)\). --- 📌 Must Remember \(\pi1(S^1)\cong\mathbb{Z}\). Brouwer Fixed‑Point Theorem: any continuous \(f:D^n\to D^n\) has a fixed point. Borsuk–Ulam Theorem: any continuous \(g:S^n\to\mathbb{R}^n\) maps some antipodal pair to the same point. Hairy Ball Theorem: a non‑vanishing continuous tangent field exists on \(S^n\) iff \(n\) is odd. Euler–Poincaré characteristic: \(\chi(X)=\sum{n\ge0}(-1)^n bn\). Poincaré duality (closed oriented \(n\)-manifold): \(Hk(M)\cong H^{n-k}(M)\). Hurewicz theorem (simply connected): the first non‑trivial homotopy group \(\pik\) is isomorphic to \(Hk\). Whitehead theorem: a map inducing isomorphisms on all \(\pin\) is a homotopy equivalence. Universal Coefficient Theorem (homology): \(0\to Hn(X)\otimes G \to Hn(X;G) \to \operatorname{Tor}(H{n-1}(X),G)\to0\). Künneth formula (product spaces): \(Hn(X\times Y)\) expressed in terms of \(\{Hi(X)\}\) and \(\{Hj(Y)\}\). --- 🔄 Key Processes Compute \(\pi1\) via Van Kampen Decompose \(X = A\cup B\) with path‑connected overlap. \(\pi1(X) \cong \pi1(A) {\pi1(A\cap B)} \pi1(B)\). Cellular homology for a CW complex Build chain groups \(Cn\) as free abelian groups on \(n\)-cells. Boundary maps given by attaching maps → compute \(Hn = \ker\partialn / \operatorname{im}\partial{n+1}\). Applying the Universal Coefficient Theorem Choose coefficient group \(G\). Use known \(Hn(X)\) to get \(Hn(X;G)\) via tensor and Tor terms. Using the Künneth formula For spaces with torsion‑free homology: \(Hn(X\times Y) \cong \bigoplus{i+j=n} Hi(X)\otimes Hj(Y)\). Add Tor terms if torsion is present. Poincaré duality check Verify orientation (top‑dimensional \(Hn\) is \(\mathbb{Z}\)). Pair \(Hk\) with \(H^{n-k}\) via cap product to get isomorphism. --- 🔍 Key Comparisons Fundamental group vs. Homology groups \(\pi1\): possibly non‑abelian, captures loop composition. \(H1\): abelianization of \(\pi1\); loses information about ordering. Simplicial complex vs. CW complex Simplicial: built from simplices, stricter combinatorial structure. CW: cells attached via continuous maps; often fewer cells, better for homotopy. Homology vs. Cohomology Homology: “chains → holes”. Cohomology: “cochains → functions on chains”; carries cup product → richer algebraic structure. Betti number vs. Torsion part Betti number \(bn\): rank (free part) of \(Hn\). Torsion: finite cyclic summands, invisible to Euler characteristic. --- ⚠️ Common Misunderstandings “Homology groups are always free” – false; many spaces have torsion (e.g., \(\mathbb{R}P^2\) has \(H1=\mathbb{Z}2\)). “\(\pi1\) = \(H1\)” – only true after abelianization; non‑abelian information is lost. “All manifolds are orientable” – not; non‑orientable manifolds have top homology \(0\) (e.g., Möbius strip). “A CW complex is the same as a simplicial complex” – they are distinct categories; a CW complex need not have simplicial structure. --- 🧠 Mental Models / Intuition Loops ↔ Strings: \(\pi1\) tracks how you can tie a string in the space; homology counts how many “holes” a string can wrap around without being contractible. Cellular chain complex: Think of each cell as a Lego brick; the boundary map tells you how bricks are glued together. Homology measures “orphan” bricks that don’t cancel. Duality: Picture a manifold as a rubber sheet; cutting a \(k\)-dimensional hole leaves a complementary \((n-k)\)-dimensional “cavity” – that’s Poincaré duality. --- 🚩 Exceptions & Edge Cases Poincaré duality fails for non‑oriented manifolds (top homology = 0). Künneth formula requires careful Tor terms when torsion is present; ignoring them yields wrong groups. Universal coefficient theorem for cohomology uses \(\operatorname{Ext}\) instead of \(\operatorname{Tor}\). Whitehead theorem requires all homotopy groups, not just \(\pi1\). --- 📍 When to Use Which Compute \(\pi1\) → use Van Kampen when space splits nicely into overlapping path‑connected pieces. Find Betti numbers → use cellular homology on a CW decomposition; easier than singular homology. Compare two spaces → check homology groups first (easier) and then \(\pi1\) if more discrimination needed. Product spaces → apply Künneth formula rather than recomputing from scratch. Manifolds → use Poincaré duality to translate a hard homology problem into a (often) easier cohomology one. --- 👀 Patterns to Recognize Exact sequence “… → A → B → C → …” → if two consecutive maps are zero, the middle group splits as direct sum of kernel and image. Euler characteristic alternating sum → a quick sanity check for computed Betti numbers. Odd vs. even dimensions → many theorems (hairy ball, antipodal maps) hinge on parity of \(n\). Non‑trivial \(\pi1\) ⇒ possible torsion in \(H1\) – look for covering space arguments (e.g., \(\mathbb{R}P^n\)). --- 🗂️ Exam Traps Choosing “homology” when the question asks for a non‑abelian invariant – the answer is \(\pi1\) or higher homotopy groups. Confusing “orientable” with “simply connected” – a torus is orientable but not simply connected. Ignoring torsion in Künneth – a product with \(\mathbb{R}P^2\) will introduce \(\mathbb{Z}2\) Tor terms; omitting them yields a wrong rank. Assuming all CW complexes have a unique cell structure – different CW structures can give different chain complexes but the same homology; the answer often lies in the homology, not the specific cells. Brouwer vs. Borsuk–Ulam – Brouwer guarantees a fixed point on a disk; Borsuk–Ulam guarantees an antipodal pair on a sphere. Mixing them leads to incorrect “must‑have‑fixed‑point” statements for spheres. ---