References:
- Munkres [@munkres_2018]
- Hatcher [@hatcher_2002]
Some fun resources:
Notation
- All maps between spaces are assumed continuous!
- All maps are pointed, i.e. we implicitly work in the category \({\mathsf{Top}}_*\).
- If left unspecified, homology is always taken with \({\mathbf{Z}}{\hbox{-}}\)coefficients.
| Notation | Definition |
|---|---|
| \(X\times Y, \prod_{j\in J} X_j, X^{\times n}\) | Direct Products |
| \(X\oplus Y, \bigoplus_{j\in J} X_j, X^{\oplus n}\) | Direct sums |
| \(X\otimes Y, \bigotimes_{j\in J} X_j, X^{\otimes n}\) | Tensor products |
| \(X\ast Y, \ast_{j\in J} X_j, X^{\ast n}\) | Free products |
| \({\mathbf{Z}}^n\) | The free abelian group of rank \(n\) |
| \(F_n, {\mathbf{Z}}^{\ast n}\) | The free group on \(n\) generators |
| \(\pi_0(X)\) | The set of path components of \(X\) |
| \(G=1\) | The trivial abelian group |
| \(G=0\) | The trivial nonabelian group |
I use \(e_G\) or \(1_G, 0_G\) to denote identity elements in a group \(G\).
\(A\times B\) denotes the direct product of modules. \(A\oplus B\) denotes a direct sum: the subset of \(A\times B\) where only finitely many terms are nonzero. Both the product and direct sum have coordinate-wise operations. For finite index sets \({\left\lvert {J} \right\rvert}< \infty\), the direct sum and product coincide, but in general there is only an injection \(\bigoplus_j X_i \hookrightarrow\prod_j X_j\). In the direct sum \(\bigoplus_j X_j\) have only finitely many nonzero entries, while the product allows infinitely many nonzero entries. So in general, I always use the product notation.
The free group on \(n\) generators is the free product of \(n\) free abelian groups, but is not generally abelian! So we use multiplicative notation, and elements \begin{align*} x \in {\mathbf{Z}}^{\ast n} = \left< a_1, \ldots, a_n\right> \end{align*} are finite words in the noncommuting symbols \(a_i^k\) for \(k\in {\mathbf{Z}}\). E.g. an element may look like \begin{align*} x = a_1^2 a_2^4 a_1 a_2^{-2} .\end{align*}
The free abelian group of rank \(n\) is the abelianization of \({\mathbf{Z}}^{\ast n}\), and its elements are characterized by \begin{align*} x\in {\mathbf{Z}}^{\ast n} = \left\langle{ a_1, \cdots, a_n }\right\rangle \implies x = \sum_n c_i a_i \text{ for some } c_i \in {\mathbf{Z}} \end{align*} where the \(a_i\) are some generating set of \(n\) elements and we used additive notation since the group is abelian. E.g. such an element may look like \begin{align*} x = 2a_1 + 4a_2 + a_1 - a_2 = 3a_1 + 3a_2 .\end{align*}
Spaces are assumed to be connected and path connected, so \(\pi_0(X) = H_0(X) = {\mathbf{Z}}\). So I virtually never consider anything occurring at index zero in these notes.
Graded objects such as \(\pi_*, H_*, H^*\) are sometimes represented as lists, which always start indexing at 1. Examples: \begin{align*} \pi_*(X) &= [\pi_1(X), \pi_2(X), \pi_3(X), \cdots] \\ H_*(X) &= [H_1(X), H_2(X), H_3(X), \cdots] .\end{align*}
Background Algebra
An injective group morphism \(f:X\hookrightarrow Y\) where \(X\) is trivial forces \(Y\) to be trivial.
There are no nontrivial homomorphisms from finite groups into free groups. In particular, any group morphism \(f: {\mathbf{Z}}_n \to {\mathbf{Z}}\) is trivial.
Let \(f: G\to H\), then \(f(1_G) = 1_H\). Supposing \(g\in G\) is torsion of order \(n\), we have \begin{align*} 1_H = f(1_G) = f(g^n) = f(g)^n ,\end{align*}
so \(f(g)\) is torsion of order dividing \(n\). But a free group is torsionfree.
This is especially useful if you have some \(f: A\to B\) and you look at the induced homomorphism \(f_*: \pi_1(A) \to\pi_1(B)\). If the former is finite and the latter contains a copy of \({\mathbf{Z}}\), then \(f_*\) has to be the trivial map \(f_*([\alpha]) = e \in \pi_1(B)\) for every \([\alpha] \in \pi_1(A)\). You can play a similar game when you take homology or cohomology.