Greg Friedman
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What (if anything) happened to Intersection Homology?
34 votes

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 ...

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References for sign conventions in homological algebra
29 votes

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 ...

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Short exact sequences every mathematician should know
24 votes

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 $$...

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So what exactly are perverse sheaves anyway?
16 votes

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 ...

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Two points of view about Borel-moore homology
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16 votes

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 ...

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Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology
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14 votes

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) ...

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Geometric meaning of L-genus
12 votes

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: ...

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Singular Homology/Cohomology as a derived functor?
12 votes

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 ...

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Which journals publish expository work?
10 votes

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 ...

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Exotic smooth structures on Lie groups?
9 votes

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 ...

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Singular analog of cellular homology
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8 votes

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 ...

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Stratified Pseudomanifold
8 votes

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 ...

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Is intersection homology the usual homology of something else?
8 votes

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 ...

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Computing Massey products via intersection theory
7 votes

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 ...

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What are examples when the equality of some invariants is good enough in algebraic topology?
7 votes

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 ...

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worked out examples in borel-moore homology
7 votes

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 ...

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When is a finite cw-complex a compact topological manifold?
7 votes

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 (...

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Cobordism categories that don't involve manifolds
7 votes

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 ...

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If $H_i(U_j)=0$ for infinitely many $j$ then $H_i(X)=0$
6 votes

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.

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How to deal with an advisor that offers you nearly no advising at all?
6 votes

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. ...

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A recommendation for a book on perverse sheaves
6 votes

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

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"Correct" definition of stratified spaces and reference for constructible sheaves?
6 votes

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 ...

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Are there non-contractible spaces $A$ and $B$ such that $A \wedge B$ is contractible?
6 votes

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\...

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Modern source for spectra (including ring spectra)
6 votes

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 ...

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references / general idea of kervaire invariant problem
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6 votes

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)

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Is there a reference containing standard mathematical notations?
6 votes

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 ...

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Books in advanced differential topology
5 votes

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 ...

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Confusion about a proof from Goresky and MacPherson's "Intersection Homology II"
Accepted answer
5 votes

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 ...

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Is singular barycentric subdivision injective?
5 votes

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 ...

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Integer cohomology of the Grassman manifold of n planes in $R^\infty$
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5 votes

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–...

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