Intersection homology is alive and well in a large number of guises. It's true that a lot of the work trended to algebraic geometry, representation theory, and categorical constructions, such as ...

This is much more of a comment than an answer, but I ran out of room in the comment box: Tyler, my new standard reference for sign issues is YOUR paper, which I post here for the benefit of others ...

How about the short exact sequence that expresses that every group can be expressed in terms of generators and relators? For any group $G$, there is a short exact sequence (in fact many) of the form $$...

If you're looking for a more geometric interpretation of perverse sheaves, you might be interested in MacPherson's 1990 lecture notes "Intersection Homology and Perverse Sheaves." As far as I know, I ...

I'll have more time to write and provide a more thorough answer later, but I think the most straightforward proof (which I agree is hard to find) comes via sheaf theory: On the one hand, there is a ...

The proof of the existence of good covers is contained in Bott & Tu on pages 42-43, though that proof does refer out to Spivak (see below). In general, if your main goal is to study (algebraic) ...

Hirzebruch himself has a very nice paper explaining (if I remember correctly) how he came up with the signature theorem and why the formulas arise in a fairly reasonable way. Here's the reference: ...

Elaborating a bit on Qiaochu Yuan's comment, if X is "nice enough" and $\mathcal{R}$ is the constant sheaf with stalks in $R$, then the singular cohomology agrees with the derived functor of the ...

There's an online Mexican journal Morfismos that seems to specialize mainly in surveys. It seems to be a small-ish operation, but they've published expository papers by well-known authors such as ...

According to the introduction to the following paper of Farrell and Jones, if $n>4$ and $\Sigma^n$ is any exotic homotopy sphere, then $T^n\#\Sigma^n$ is not diffeomorphic to $T^n$. So lots of ...

Just thinking on my feet: From what I can tell, the trouble with setting up a "singular cell" version of homology is that simplices, whether part of a simplicial complex or on their own, have ...

It's not! Okay, that's a little too glib. You're right that it's often required that $X^{n-1}=X^{n-2}$, but it depends somewhat on your purpose. In most of my recent papers, including those I'm ...

The short answer is "no". But you might be interested in some work Markus Banagl is doing along these lines. He has a functor that assigns to a space X an "intersection space" $I^{\bar p}X$. Then ...

While not quite an answer from the perspective you're thinking of, one approach to this (at least if you're in a smooth manifold), would be to use a model of cohomology in which cocycles are ...

Maybe this is a little too on the nose, but: obstruction theory to extension and lifting problems in the sense of, e.g., Whitehead Chapter 6 or Davis-Kirk Chapter 7. Here vanishing of the obstruction ...

While not so easy to work out directly, on "nice enough" spaces, Borel-Moore homology can be interpreted as what you get by allowing singular chains that are formal linear combinations of a possibly ...

In a certain sense, this is one of the starting questions for surgery theory, which aims to classify all manifolds, to the extent possible. You can get some idea of the complexity of this problem (...

One famous (in my field!) example is Witt space bordism. Witt spaces are not manifolds but rather pseudomanifolds (which aren't so far off from manifolds, but they can have singularities). A ...

This is true under some mild assumptions using the relationship between unions of spaces and direct limits of homology groups. See Proposition 3.33 of Hatcher's Algebraic Topology.

The following suggestion may be naive, and I would say it definitely depends a lot on the personalities involved, so you should consider how things might play out before taking the following advice. ...

Here are two possibilities: Topological Invariants of Stratified Spaces by Markus Banagl Intersection Homology & Perverse Sheaves: with Applications to Singularities by Laurenţiu G. Maxim

A bit more reader friendly than Kashiwara-Schapira is Borel's Intersection Cohomology. It doesn't treat perverse sheaves, but it has a good overview of stratified spaces in the first few chapter and ...

I just saw this in Ravenel's "orange book"! Let $X$ be any simply connected CW complex whose reduced homology is all torsion. Let $X_{(p)}$ be the p-localization for a prime $p$. Then for primes $p\...

Probably neither of these will be exactly what you're looking for, but here are two references that come to mind and might have some of what you want: Algebraic Topology by Robert M. Switzer is a ...

Here's a recent survey article by Victor Snaith: http://chucha.math.cinvestav.mx/morfismos/v13n2/arfsurveyMFMS.pdf (I think there's also a copy on the arxiv)

I'm not sure this is quite what you have in mind, but there is a "comprehensive" LaTeX symbol list: http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf Unfortunately, it doesn't ...

Avoiding suggestions already made: Guillemin and Pollack, Differential Topology, is a classic. You can also find pieces of a lot of these things in books that are a bit broader, for example: Topology ...

As noted by Chris Gerig in the comments, letting $cX$ denote the open cone on the compact space $X$ then $(cX)\times (cY)\cong c(X*Y)$, where $X*Y$ is the join. In the case at hand, Goresky and ...

No: consider the circle. Let $\sigma_1$ be the singular 1-simplex that wraps exactly once counterclockwise around the circle, and let $\sigma_2$ be obtained from $\sigma_1$ by reversing the ...

I don't know if these have everything that you want, but see the following: Brown, Edgar H., Jr. The cohomology of BSOn and BOn with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–...