Topology 2
Main reference book:
Algebraic Topology by Hatcher.
Syllabus: Chapter 2, 3.
Lecture 9: Excision II
- Proof of existence of \(\rho: C_{\bullet}(X) \to C_{\bullet}^{\sU}(X) \) which gives a homotopy inverse of the inclusion \(i\).
Lecture 8: Excision I
- \({\mathcal U}\)-small chains. Alternative way of stating excision.
- Linear chains in a convex subset of \({\mathbb R}^n\), barycentric subdivision.
- General barycentric subdivision operator.
Lecture 7: Long exact sequence II
- Proof of the long exact sequence theorem.
- Definition of reduced homology and its relation with non-reduced homology. Calculation of homology of spheres.
- Statement of Excision theorem.
Lecture 6: Long exact sequence I
- Calculation of simplicial homology of torus and real projective space.
- short exact sequence of complexes \( \Rightarrow \) long exact sequence of homology groups : definition of
connecting homomorphism.
Assignment 1:
- Calculate the simplicial homology of klein bottle and \(S^2\).
- State homotopy invariance theorem for singular homology. Calculate \(H_i({\mathbb R}^n)\) for \(i\in {\mathbb Z}\).
- Show that if \(X\) is the disjoint union of its path connected components \(X_{\alpha}\), then
$$ H_n(X)= \oplus_{\alpha} H_n(X_{\alpha}) $$
Lecture 5: Simplicial homology
- Definition of Delta complexes.
- Examples of Delta complexes, \(S^1\), Torus, \({\mathbb R}P^2\), Klein bottle.
- Definition of simplicial homology.
- Calculation of simplicial homology of \(S^1\).
Lecture 4: Homotopy invariance of homology - II
- Proof of homotopy invariance. Definition of homotopy \(P\) from \(C_n(X) \to C_{n+1}(Y)\) given a homotopy of maps from \(X\) to \(Y\).
$$ \partial P + P \partial = g_{\#}-f_{\#} $$
- Definition of chain homotopy of morphism of chain complexes. Exercise: Show chain homotopy is an equivalence relation and
is well behaved w.r.t. pre and post composition of maps.
Lecture 3: Homotopy invariance of homology - I
- Continuous maps induce map on singular homology.
- Notation \([x]\) for classes in homology. If \(f_{\bullet}\) is a morphism of chain complexes, then
$$ f_*[x]= [f(x)] $$ where \(x \in {\rm Ker}(d_n)\).
- Statement of homotopy invariance. Its (idea of) proof in the case of \(H_0\) and for the case of homology class special loops.
Lecture 2 : Definition of singular homology
- Definition of chain complex, morphism between chain complexes, homology of a chain complex.
- singular chain complex of a topological space, singular homology
- \(H_0(X)\cong {\mathbb Z}\) if \(X\) is non empty and path connected.
- Homology of a point.
Reading exercise: Introduction of chapter 2, section on singular homology.
Lecture 1 : Singular simplices
- Brief overview of Singular homology (as a functor and homotopy invariance). Difference with higher homotopy groups.
- Definition of standard \(n\) simplex and singular simplices.
- Uniqueness and existence of piecewise linear maps from convex hull of finitely many points to \({\mathbb R}^n\)
- Definition of \(C_n(X)\) and the boundary map \(C_n(X)\xrightarrow{\partial}C_{n-1}(X)\)
Exercise: Let \(C\) be the convex hull of \(n\) points \(\{v_1,...,v_n\} \) in a Euclidean space in general position.
Show that given any \(\{w_1,..,w_n\} \subset {\mathbb R}^m\) there exists a unique piecewise linear map from \(C \to {\mathbb R}^m\)
sending \( v_i \mapsto w_i\).