Does anyone know a good description of homotopy classes of continuous functions $\Sigma_g \longrightarrow \mathbb{R}P^2$, where $\Sigma_g$ is the closed oriented surface of genus $g > 1$.
Thanks for the attention!
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1$\begingroup$ See the paper "Homotopy classification the JHC Whitehead way" by G. Ellis, Exposition. Math. 6 (2) (1988) 97–110. available at http://.groupoids.org.uk/pdffiles/Ellis-homclass.pdf . $\endgroup$– Ronnie BrownMar 5, 2017 at 17:33
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$\begingroup$ @RonnieBrown Sorry, but your link is broken. It seems this works. $\endgroup$– Sebastian GoetteMar 5, 2017 at 18:55
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2 Answers
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This mathstackexchange answer gives a nice description of the set of homotopy class of maps from $T^2=\Sigma_1$ to any space $X$, and includes a particular mention of the example $X={\mathbb R}P^2$. Answers to this mathoverflow question give a couple of other ways to calculate the set $[T^2, {\mathbb R}P^2]$. It seems to me that all these methods can be extended to general orientable surfaces. The answer is as follows: there are ${\mathbb N}$ homotopy classes of maps that induce zero on $H_1$, plus $(2^{2g}-1)\cdot 2$ maps that are not zero on $H_1$ (there are two homotopy classes for each possible non-zero homomorphism on $H_1$).
answered Mar 5, 2017 at 19:42
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$\begingroup$ You are right of course. I was wondering if a map to non-orientable surface can have an integer-valued degree, but at least for $S^2$ this is possible. Maybe I should have read Ellis' paper more carefully. $\endgroup$ Mar 5, 2017 at 19:44
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$\begingroup$ Thank you very much for all these reference I found everything I need in it :) $\endgroup$– Selim GMar 6, 2017 at 8:29
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$\begingroup$ I mention that in order to show an idea of available methods, Example 12.3.13 of the book "Nonabelian Algebraic Topology", advertised at groupoids.org.uk/nonab-a-t.html , gives an example of calculating some homotopy classes of maps $K \to Y$ where $K$ is the product of two copies of $\mathbb RP^2$ and $Y$ is $\mathbb RP^3$ with homotopy groups killed above dimension 3. $\endgroup$ Mar 7, 2017 at 15:16
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Perhaps the following article might be useful. It is written by Daciberg L. Gonçalves and Mauro Spreafico, with the title "THE FUNDAMENTAL GROUP OF THE SPACE OF MAPS FROM A SURFACE INTO THE PROJECTIVE PLANE".