# Oriented finite CW complex with prescribed homology?

Moore spaces are finite CW complexes with prescribed homology, but they may be non-orientable and even not topological manifolds. Are there oriented, connected CW complexes with prescribed homology (with $\mathbb{Z}$-coefficents?) If such a complex is also a topological manifold then there are additional restrictions (Poincare duality) so we can't expect such spaces to be homotopic to topological manifolds.

As a side note: Should I consider CW complexes as the "most general & nice" class of topological space that people commonly work with upto homotopy. (This is really a Yes/No question intended to be interpreted as basic. For example, the class of CW complexes (upto homotopy) includes all other classes of topological spaces I know: topological manifolds (with or without boundary), simplicial complexes, delta complexes. In fact, except the class of topological manifolds (with or without boundary), all of these describe the same class of topological spaces upto homotopy) For me, topological manifolds should have a countable basis.

• What's an "oriented CW-complex"? – Fernando Muro Dec 26 '12 at 15:50
• Is there no notion of "orientable" for CW complexes? I naively assumed that such a thing made sense for CW complexes. Does it only make sense to ask if a topological space is "orientable" if it is a topological manifold. – LMN Dec 26 '12 at 16:44
• When you deal with "nice" singular spaces as pseudomanifolds, orientation makes sense. Because in the case of pseudomanifolds, you have a top stratum (the regular part) that is a manifold and this stratum is dense. Hence orientation in that case is orientation of the regular part, orientable pseudomanifolds of dimension $n$ have a fundamental class in top dimensional homology. – David C Dec 26 '12 at 17:16
• It can make sense for homology manifolds too: en.wikipedia.org/wiki/Homology_manifold – Ian Agol Dec 27 '12 at 7:15
• Orientation can make sense for homology manifolds too: en.wikipedia.org/wiki/Homology_manifold But these will satisfy a version of Poincare duality. – Ian Agol Dec 27 '12 at 7:17

To answer the question of your side note. CW-complexes are not enough, there are needs to work with "convenient categories of topological spaces" where you have all CW-complexes, function spaces, you also want your category to be cartesian closed... This is very important when you want to study iterated loop spaces (see Peter May "Geometry of iterated loop spaces"), bar constructions, classifying spaces...

You can also have at a look at: Norman Steenrod, "A convenient category of topological spaces", Michigan Math. J. 14 (1967) 133–152 or Neil Strickland, The Category of CGWH Spaces. available on his homepage.

The introduction of N. Steenrod's paper is very good and the paper is worth reading.

Ronnie Brown has done some important works on these "convenient categories":

Ronnie Brown, Some problems of algebraic topology: a study of function spaces, function complexes, and FD-complexes, DPhil thesis (part A), Oxford University, 1961

Ronnie Brown, Ten topologies for X×Y, Quart. J.Math. (2) 14 (1963), 303–319.

Ronnie Brown, Function spaces and product topologies, Quart. J. Math. (2) 15 (1964), 238–250.

• Just for the record, the question of a category "adequate and convenient for all purposes of topology" was put in the paper "Ten topologies..." and a full list of those properties, but without the term "cartesian closed" was given in the paper "Function spaces...", whose results are stronger than those in Steenrod's paper. I also sent him a copy of my thesis. I do not know why Steenrod did not explain all this in his paper. Very curious! – Ronnie Brown Dec 26 '12 at 22:29
• The above listed papers and Part I of the thesis are available from pages.bangor.ac.uk/~mas010/publicfull.htm – Ronnie Brown Dec 29 '12 at 11:11