# Topology of manifolds

Where can I find a stricter presentation of topology of manifolds, then in section 0.4 in Griffiths-Harris? For example, they define the map $H_k \times H_{n-k}$ by presenting a cycle by a submanifold and intersection form. But it's known, that not all homology classes are representable by a submanifold. It can contain singularities in codimension 2. Griffiths and Harris wtite nothing about it. But in the classical topology all is defined strictly, but it's very difficult to calculate anything, using their definition (for example, in Hatcher, Algebraic topology). Then I want to justify the approach of Griffiths-Harris. Also they use, that we can compute homology groups buy smooth simplexes or using triangulations and simplexes, contained in triangulation. The fact of existence of triangulation, as far as I know, is quite difficult, and I don't want to use it without proof. By the wyay, Griffiths and Harris writes, that their book is quite self-contained.

• As I was once reminded on this site before, the approppriate tag for such questions is "differential-topology", not "differential-geometry". – Piotr Pstrągowski Mar 3 '13 at 14:17