Convenient maps from one-relator model two-complexes into the real projective plane
Słowa kluczowe
Convenient maps, strong surjections, two-dimensional complexes, aspherical complexes, cohomology with local coefficientsAbstrakt
Let $f$ be a map from a one-relator model two-complex $K_{\mathcal{P}}$ into the real projective plane. The composition $\varrho\circ f_{\#}$ of the homomorphism $f_{\#}$ induced by $f$ on fundamental groups with the action $\varrho$ of $\pi_1(\mathbb{R}\mathrm{P}^2)$ over $\pi_2(\mathbb{R}\mathrm{P}^2)$ provides a local integer coefficient system $f_{\#}^{\varrho}$ over $K_{\mathcal{P}}$. We prove that if the twisted integer cohomology group $H^2(K_{\mathcal{P}};_{f_{\#}^{\varrho}}\!\mathbb Z)=0$, then $f$ is homotopic to a non-surjective map. As an intermediary step for the proof, we show that if $H^2(K_{\mathcal{P}};_{\beta}\!\mathbb Z)=0$ for some local integer coefficient system $\beta$ over $K_{\mathcal{P}}$, then $K_{\mathcal{P}}$ is aspherical.Bibliografia
C. Aniz, Strong surjectivity of mappings of some 3-complexes into 3-manifolds, Fund. Math. 192 (2006), 195–214.
C. Aniz, Strong surjectivity of mappings of some 3-complexes into MQ8 , Cent. Eur. J. Math. 6 (2008), no. 4, 497–503.
W.A. Bogley and S.J. Pride, Calculating Generators of Π2 , Two-Dimensional Homotopy and Combinatorial Group Theory (C. Hog-Angeloni, W. Metzler and A.J. Sieradski, eds.), 157–188, Cambridge University Press, 1993.
M.C. Fenille, Strong surjections from two-complexes with trivial top-cohomology onto the torus, Topol. Appl. 210 (2016), 63–69.
M.C. Fenille, The trivial homotopy classe of maps from two-complexes into the real projective plane, Topol. Methods Nonlinear Anal. 46 (2015), 603-615.
M.C. Fenille and O.M. Neto, Root problem for convenient maps, Topol. Methods Nonlinear Anal. 36 (2010), no. 2, 327–352.
M.C. Fenille and O.M. Neto, Strong surjectivity of maps from 2-complexes into the 2-sphere, Cent. Eur. J. Math. 8 (2010), no. 3, 421–429.
J. Huebschmann, Aspherical 2-complexes and an unsettled proble of J.H.C. Whitehead, Math. Ann. 258 (1981), 17–37.
R.C. Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. (2) 52 (1950), 650–655.
A.J. Sieradski, Algebraic topology for two-dimensional complexes, Two-Dimensional Homotopy and Combinatorial Group Theory (C. Hog-Angeloni, W. Metzler and A.J. Sieradski, eds.), Cambridge University Press, 1993, 51–96.
G.W. Whitehead, Elements of Homotopy Theory, Springer–Verlag New York Inc., 1978.
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