References:
Further reading:
Notation
(Conventions)
- Maps between spaces are continuous.
- Maps of spaces are pointed unless stated otherwise, that is, they are morphisms in \(\Top_*\).
- Homology and cohomology have \(\ZZ\) coefficients unless stated otherwise.
| Notation | Definition |
|---|---|
| \(X\cross Y, \prod_{j\in J} X_j, X^{\cross n}\) | Direct products |
| \(X\oplus Y, \bigoplus_{j\in J} X_j, X^{\oplus n}\) | Direct sums |
| \(X\tensor Y, \bigotimes_{j\in J} X_j, X^{\tensor n}\) | Tensor products |
| \(X\ast Y, \ast_{j\in J} X_j, X^{\ast n}\) | Free products |
| \(\ZZ^n\) | The free abelian group of rank \(n\) |
| \(F_n, \ZZ^{\ast n}\) | The free group on \(n\) generators |
| \(\pi_0(X)\) | The set of path components of \(X\) |
| \(G=1\) | The trivial group, written multiplicatively |
| \(G=0\) | The trivial group, written additively |
The identity element of a group \(G\) is written \(e_G\), \(1_G\), or \(0_G\).
(Direct sums and direct products)
For a family of modules \((X_j)_{j\in J}\), the direct product \(\prod_{j\in J} X_j\) is the set of all tuples \((x_j)_{j\in J}\) with coordinatewise operations, and the direct sum \(\bigoplus_{j\in J} X_j\) is the submodule of tuples with only finitely many nonzero entries. The inclusion \(\bigoplus_{j\in J} X_j \injects \prod_{j\in J} X_j\) is an isomorphism when \(J\) is finite, and it is not surjective when infinitely many \(X_j\) are nonzero.
(Free groups and free products)
The free group on \(n\) generators is the free product of \(n\) copies of \(\ZZ\), and it is nonabelian for \(n\geq 2\). It is written multiplicatively: elements of \[ \ZZ^{\ast n} = \gens{a_1, \ldots, a_n} \] are reduced finite words in the symbols \(a_i^k\) for \(k\in \ZZ\), such as \[ x = a_1^2 a_2^4 a_1 a_2^{-2}. \]
(Free abelian groups)
The free abelian group \(\ZZ^n\) of rank \(n\) is the abelianization of \(\ZZ^{\ast n}\), written additively. For a basis \(a_1,\ldots,a_n\), each \(x\in\ZZ^n\) is uniquely \(x = \sum_{i=1}^n c_i a_i\) with \(c_i \in \ZZ\); for example, \[ 2a_1 + 4a_2 + a_1 - a_2 = 3a_1 + 3a_2. \]
(Indexing conventions and list notation)
For path-connected \(X\), \(\pi_0(X)\) is a point and \(H_0(X) \cong \ZZ\), and a graded object such as \(\pi_*\), \(H_*\), or \(H^*\) written as a list starts in degree \(1\): \[ \begin{aligned} \pi_*(X) &= [\pi_1(X), \pi_2(X), \pi_3(X), \ldots], \\ H_*(X) &= [H_1(X), H_2(X), H_3(X), \ldots]. \end{aligned} \]
Background algebra
If \(f\colon X\injects Y\) is an injective group homomorphism and \(Y\) is trivial, then \(X\) is trivial.
Homomorphisms from finite groups to free groups (Tag PR-WB7MM)
Every homomorphism from a finite group to a free group is trivial, since free groups are torsion-free. In particular, every homomorphism \(\ZZ/n \to \ZZ\) is trivial.
Let \(f\colon G\to H\) be a homomorphism from a finite group to a free group, so \(f(1_G) = 1_H\). Each \(g\in G\) has finite order \(n\), and \[ 1_H = f(1_G) = f(g^n) = f(g)^n, \] so \(f(g)\) has finite order dividing \(n\). Free groups are torsion-free, so \(f(g) = 1_H\).
The same argument shows that every homomorphism from a finite group to a torsion-free group is trivial. For a continuous map \(f\colon A\to B\) with \(\pi_1(A)\) finite and \(\pi_1(B)\) torsion-free, such as \(\pi_1(B)\cong\ZZ^n\) or a free group, the induced homomorphism \(f_*\colon \pi_1(A) \to \pi_1(B)\) is trivial. Likewise, if \(H_k(A)\) is finite and \(H_k(B)\) is torsion-free, then \(f_*\colon H_k(A)\to H_k(B)\) is zero, and if \(H^k(B)\) is finite and \(H^k(A)\) is torsion-free, then \(f^*\colon H^k(B)\to H^k(A)\) is zero.