# Hopf's 1935 paper “About maps from spheres to spheres of lower dimension”

I've been trying to find Hopf, Heinz (1935), "Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension", Fundamenta Mathematicae (Warsaw: Polish Acad. Sci.) 25: 427–440, ISSN 0016-2736 but have had no luck so far.

I want to know what he did in that paper. I can read German so if someone would point me to where I can download it I'd be very grateful.

I did a thorough search and it might simply not be available. Hence, if that's the case, can anyone tell me what he did in that paper? Thanks a lot.

-
According to my experience, if a journal has its own Wikipedia article and if there is some online archive where older issues are available, a link usually can be found in the Wikipedia article. This is true for Fundamenta mathematcae, too. (Although in this case the link provided on Wiki is a little outdated, after a few clicks you will be on the search page mentioned in t.b.'s answer.) –  Martin Sleziak Sep 8 '12 at 11:46

The home page of the Polish Virtual Library has a good search interface and the older issues of the classical Polish journals such as Fund. Math., Studia Math., etc. as well as the monograph series are available for free more or less in their entirety. Here's a link to the old repository which is easy to browse, but no longer updated.

Here's a direct link to Hopf's paper you're looking for and here's the link to the library's details on the article.

Here's the freely available Zentralblatt entry containing a summary of the paper (in German).

A few generic remarks on locating papers online:

As a rule, when looking for a specific published paper, I just search for the name of the journal which quickly leads me to the journal's home page. The article (if available) usually is only a few clicks away (modulo subscription). Looking for author and title is less likely to lead to the paper (especially if it is old and famous) because you're more often served papers that cite the paper than the paper itself. Of course, if the paper is relatively new (newer than mid-90ies, say), chances are that the paper can be found somewhere on the author's homepage or on the arXiv, see also the UC Davis front end of arXiv.

For papers that were published after around 1940 MathSciNet often leads to the paper using the “article” or “journal” link it provides. Unfortunately, MathSciNet it isn't fully linked with the journals apart from those of the major publishers and it needs a subscription for searching and viewing the entries, but most universities (or at least the libraries) do have access.

Alternatively, you can use the Zentralblatt which also works (in a rather limited way) without subscription, but if you have full bibliographical information you can find the papers and their reviews there.

-
I tried to reproduce your search-find-path. How did you get to Polish vVirtual Library from Fundamenta Mathematicae? I'm asking because if I search the former for "Hopf Sphären" it turns up nothing. –  Rudy the Reindeer Sep 7 '12 at 15:34
Click on "all issues", scroll down to the bottom, click on Polish Virtual Library here (third line in the screen shot). Not that easy to find, admittedly :) –  t.b. Sep 7 '12 at 15:41
Got it. Now I feel less stupid for not being able to find it on my own. Plus I'm starting to think that you have the force. : ) –  Rudy the Reindeer Sep 7 '12 at 15:46
Well, many of the classic papers of my favorite subject are to be found on that page, so at some point (after quite a bit of exasperation) I learned how to navigate that particular page... –  t.b. Sep 7 '12 at 15:53

Collected Papers (ISBN: 3540571388) has the original German ("Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension" p.471) and an English Translation ("On Mapping Spheres onto Spheres of Lower Dimension" p.485).

Here seems to be an English translation.

-
Thank you! That is very helpful, I was completely unaware that I could order his works nicely bound into a book. I have added it to my want to read list : ) –  Rudy the Reindeer Feb 18 '13 at 19:17

To summarise the contents of the paper:

The paper is a generalisation of his earlier paper from 1931 where he exhibited a surjective continuous map $S^3 \to S^2$. In his 1935 he generalises this to the following theorem:

For all $k \geq 1$ there exists a continuous surjective map $S^{4k - 1} \to S^{2k}$.

As a direct consequence we get that $\pi_{4k - 1} (S^{2k}) \neq \{ 0 \}$ for all $k \geq 1$.

-