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A chain map inducing isomorphisms on all homology groups. The same definition applies to a map of cochain complexes using cohomology groups. A quasi-isomorphism need not be an isomorphism in each degree.

For a compact Riemann surface with a symplectic basis $a_i,b_i$ of its first homology group, cutting along that basis and integrating a primitive givesfor closed smooth one-forms. For a holomorphic differential form $\omega$ and a differential of the third kind $\eta$, the corresponding boundary calculation givesThe paths and representatives of the cycles are chosen in the same cut surface. Normalized holomorphic differential forms consequently have a symmetric period matrix with positive definite imaginary part.

For $0\le k<m$, collapse the standard skeleton of a CW complex ${\mathbb{RP}}^k$ inside Real projective space ${\mathbb{RP}}^m$. The quotient has one base point and one topological cell in each dimension $k+1,\dots ,m$.

Between two surviving consecutive cells, the integral cellular boundary is multiplication by two in even source dimension and zero in odd source dimension. The reduced cellular boundary out of the bottom surviving cell is zero, since the lower cellular chain group has been collapsed. This exception must be applied before computing homology.

For example, ${\mathbb{RP}}^4/{\mathbb{RP}}^1$ has cells in dimensions two, three and four, with reduced chain sequenceThus its nonzero reduced integral homology groups are ${\tilde{H}}_2=\mathbb{Z}$ and ${\tilde{H}}_3=\mathbb{Z}/2$. Steenrod squares on the relative mod-two cohomology are inherited from Real projective space. If $k=m$, the quotient is a single point.

For a continuous map $f:M\to N$ between connected closed oriented manifolds of the same positive dimension $n$, the topological degree is the integer determined byThe chosen fundamental classes fix the signs. This homology definition makes sense without differentiability and is invariant under homotopy. For spheres, it is the integer by which the induced map multiplies their top homology group.^[ref]

If $f$ is smooth and $y$ is a regular value, the finite preimage giveswhere the sign compares the chosen orientations in source and target. Thus opposite local signs can cancel: topological degree is a signed count, rather than the number of preimages.

For the circle map $z\mapsto z^k$, with $|z|=1$ and $k$ an integer, the topological degree is $k$. An orientation-reversing reflection has topological degree $-1$, while a constant map has topological degree zero. A nonzero topological degree forces surjectivity; a homotopy between continuous maps preserves their topological degree.

A homeomorphism preserves every homology group, hence the Euler characteristic wherever the finite rational Betti number sum is defined. In fact, homotopy invariance of homology makes Euler characteristic invariant under homotopy equivalence. Thus different finite CW complex structures on the same space have equal alternating cell counts.^[ref]