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Given a topological space $X$ and a non-negative integer $k$, we study the self-homotopy equivalences of $X$ that do not change maps from $X$ to $n$-sphere $S^n$ homotopically by the composition for all $n\geq k$. We denote by $\E_{k}^{\sharp}(X)$ the set of all homotopy classes of such self-homotopy equivalences. This set is a dual concept of $\E_\sharp^k(X)$, which has been studied by several authors. We prove that if $X$ is a finite CW complex, there are at most a finite number of distinguishing homotopy classes $\E_{k}^{\sharp}(X)$, whereas $\E_{\sharp}^{k}(X)$ may not be finite. Moreover, we obtain concrete computations of $\E_{k}^{\sharp}(X)$ to show that the cardinal of $\E_{k}^{\sharp}(X)$ is finite when $X$ is either a Moore space or co-Moore space by using the self-closeness numbers.
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참고문헌 (12)
참고문헌 신청
A. Sieradski / 1970 / Twisted self-homotopy equivalences / Pacific J. Math 34 : 789 ~ 802
A. Sieradski / 1972 / Stabilization of self-equivalences of the pesudoprojective spaces / Michigan. Math. J 19 : 109 ~ 119
B. Gray / 1975 / Homotopy Theory / Academic Press, Inc
H. Choi / 2014 / Certain self homotopy equivalences on wedge products on Moore spaces / Pacific J. Math 272 (1) : 35 ~ 57
H. Choi / 2015 / Certain numbers on the groups of self-homotopy equivalences / Topology Appl 181 : 104 ~ 111
A. Sieradski / 1972 / Stabilization of self-equivalences of the pesudoprojective spaces / Michigan. Math. J 19 : 109 ~ 119
B. Gray / 1975 / Homotopy Theory / Academic Press, Inc
H. Choi / 2014 / Certain self homotopy equivalences on wedge products on Moore spaces / Pacific J. Math 272 (1) : 35 ~ 57
H. Choi / 2015 / Certain numbers on the groups of self-homotopy equivalences / Topology Appl 181 : 104 ~ 111
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