5
votes
1answer
358 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
2answers
808 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 $\widetilde D = \mathbb C \setminus D$ ...
12
votes
0answers
243 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
1answer
408 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 Poincaré's Analysis Situs and Dieudonné's History of Algebraic and Differential Topology, and I haven't found the source of this ...
5
votes
1answer
356 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
0answers
419 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
4answers
940 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
1answer
567 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
0answers
277 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
1answer
383 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 ...
15
votes
0answers
866 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
0answers
308 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?
38
votes
8answers
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
1answer
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
7answers
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
0answers
447 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
2answers
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
0answers
576 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
3answers
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
2answers
473 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 ...