Re: Early intimations of categories

[Nath Rao <khazanarao@gmail.com>]

[-- Attachment #1: Type: text/plain, Size: 1926 bytes --]

On 6/11/24 18:21, Michael Barr, Prof. wrote:
[...] Although I am not sure when the homomorphism induced by a continuous map was known.
[...]

I was struck by the above sentence. I thought that surely Lefschetz fixed point index was based on the induced homomorphism. But a cursory look at the two papers of Lefschetz listed in the Wikipedia page<https://url.au.m.mimecastprotect.com/s/HfgRCQnM1Wf2Gl61TxCUzI?domain=en.wikipedia.org> on this was illuminating: The 1926 paper seems to get by without even using groups. The 1937 paper goes directly to the matrix of the induced homomorphism in rational homology, by using bases.

On the positive side, Lefschetz's 1941/42 version of the AMS Colloquium notes (vol 27, not the original 1930 version, published as vol 12; I thought that AMS made the former freely available, but I could not find it in the AMS bookstore), in VII.5.11, very briefly mentions the homomorphism in homology induced by a mapping, and in his appendix in this book (on his fixed point theorem), P. A. Smith makes use of it. It seems that people primarily used Cech and Vietoris type theories based on coverings, where defining the induced homomorphism is harder than for singular (co)homology. [Lefschetz VIII.5.11 is defined for the groups based on specific coverings.]

-Nath Rao



You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message.

View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Learn more about Microsoft 365 Groups<https://aka.ms/o365g>


[-- Attachment #2: Type: text/html, Size: 4786 bytes --]