# Tagged Questions

**4**

votes

**1**answer

347 views

### What was Seifert's contribution to the Seifert-van Kampen theorem?

The Seifert-van Kampen theorem is the classical theorem of algebraic topology that the fundamental group functor $\pi_1$ preserves pushouts; more often than not this is referred to simply as the van ...

**13**

votes

**2**answers

789 views

### Who first defined _simply connected_, reference?

The following definition is due to Donald J. Newman:
A connected open subset $D$ of the plane $\mathbb C$
is simply connected
if and only if its complement $\widetilde D = \mathbb C \setminus D$
...

**11**

votes

**0**answers

239 views

### Who stated and proved the “Hopf lemma” on bilinear maps?

If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$.
Nondegenerate here means ...

**6**

votes

**1**answer

396 views

### Where does the notation $\pi_1(X,x)$ for the fundamental group first appear?

I've spent the last half hour browsing Stillwell's translation of PoincarÃ©'s Analysis Situs and DieudonnÃ©'s History of Algebraic and Differential Topology, and I haven't found the source of this ...

**5**

votes

**1**answer

344 views

### Generalization of the Lefschetz fixed point theorem

I have encountered a certain generalization of the Lefschetz fixed point theorem as folklore, and I am hoping that someone out there knows its provenance or can otherwise refer me to a source where it ...

**9**

votes

**0**answers

416 views

### What is Quillen's contribution to index theorem?

In the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne it is said that "Our book is based on a simple principle, which we learned from D. Quillen: Dirac operators are a ...

**19**

votes

**4**answers

926 views

### The role of ANR in modern topology

Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ ...

**11**

votes

**1**answer

558 views

### Equivariant homotopy theory: some history questions

I have sometimes wondered about the following:
(1) Who was the first articulate that in dealing with $G$-equivariant cohomology theories ($G$ a finite group or a compact Lie group), it is best to ...

**10**

votes

**0**answers

273 views

### What is the history of the notion of subdivision of categories?

A recent answer by Peter May prompts me to ask a question which I have been considering to ask for several months. (The reason why I have not asked it before is that it is not directly related to my ...

**5**

votes

**1**answer

375 views

### Historical question: fiber bundles

I am sorry if this question is too trivial but I couldn't find the answer.
Who did first classify topological principal $G$-bundles for some topological group $G$? So, I mean that equivalence ...

**13**

votes

**0**answers

825 views

### Origins of the Nerve Theorem

Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question:
Who was the first to prove the Nerve Theorem?

**6**

votes

**0**answers

306 views

### Why the $M$ for Thom spaces?

I've heard $E$ is for entire space, $B$ is for base space, so what is $M$ for?

**35**

votes

**8**answers

3k views

### Natural transformations as categorical homotopies

Every text book I've ever read about Category Theory gives the definition of natural transformation as a collection of morphisms which make the well known diagrams commute.
There is another possible ...

**17**

votes

**1**answer

1k views

### The whole plethora of topology

In his answer to a recent MO question, Johannes Ebert sketches the proof of a very nice result (implying that homotopy spheres are parallelizable) which, as he says, involves the whole plethora of ...

**11**

votes

**7**answers

2k views

### History of classifying spaces

Where did the idea and formal definition of the "classifying space of a (small) category" first appear?
Added: As Andy Putman noted below, the "classical" early reference for this is G. Segal's ...

**6**

votes

**0**answers

441 views

### Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes

Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...

**28**

votes

**2**answers

2k views

### Timeline of cohomology (1935 to 1938)

There was a recent question on intuitions about sheaf cohomology, and I answered in part by suggesting the "genetic" approach (how did cohomology in general arise?). For historical material specific ...

**7**

votes

**0**answers

568 views

### Original references for the homotopy groups pi_5 of SU(3) and pi_4 of SU(2)?

For revision of a paper (http://arxiv.org/abs/1008.1189), I'd like to
correct my references to the original work on aspects of the homotopy
groups pi_5 of SU(3) and pi_4 of SU(2). I'm not a ...

**11**

votes

**3**answers

1k views

### Survey articles on homotopy groups of spheres

Are there general surveys or introductions to the homotopy groups of spheres? I'm interested especially in connections to low-dimensional geometry and topology.

**7**

votes

**2**answers

468 views

### Reference request for relative bordism coinciding with homology in low dimensions

It's a standard fact that, for finite CW complexes, the relative (edit: oriented) bordism group $\Omega_n(X,A)$ coincides with the homology $[H_\ast(X,A;\Omega_\ast(pt))]_n\simeq H_n(X,A)$ for ...