## On mod $$p$$ $$A_p$$-spaces.(English)Zbl 1373.55013

Let $$X$$ be a $$p$$-local $$A_p$$-space for an odd prime $$p$$ with cohomology ring $H^\ast(X, \mathbb{Z}/p\mathbb{Z})=\bigwedge (x_{2m_1-1}, \ldots, x_{2m_r-1}), \quad \text{deg}(x_{2m_i-1})=2m_i-1$ where $$m_1\leq m_j$$ for all $$j$$. Then by Y. Hemmi [Hiroshima Math. J. 24, No. 3, 583–605 (1994; Zbl 0865.55008)] we have that there is a space $$R_p(X)$$, called the modified projective space over $$X$$, with a map $$\varepsilon : \Sigma X \to R_p(X)$$; and in particular, in the present case its cohomology ring contains a polynomial algebra truncated at height $$p+1$$ on generators $$y_{2m_1}, \ldots, y_{2m_r}$$ satisfying the relations $$\varepsilon^\ast(y_{2m_i})=\sigma^\ast(x_{2m_i-1})$$ and equipped with a mod 3 Steenrod algebra action. Using this result the authors prove that if we set $m=\text{gcd}\{m_i \mid m_i \leq pm_1 \},$ then $$m \mid p-1$$. They find that this fact is also applicable to the classification of rank 3, mod 3 $$A_3$$-spaces (or equivalently, rank 3 3-local homotopy associative $$H$$-spaces) $$X$$. It can be actually used to detect the possible types of $$X$$ and so in addition, by using the result for the 3-projective space $$P_3(X)$$ over $$X$$ by N. Iwase [Mem. Fac. Sci., Kyushu Univ., Ser. A 38, 285–297 (1984; Zbl 0555.55003)], the authors succeed in obtaining a list of such spaces $$X$$.

### MSC:

 55P45 $$H$$-spaces and duals 55S25 $$K$$-theory operations and generalized cohomology operations in algebraic topology 55N15 Topological $$K$$-theory 55P15 Classification of homotopy type

### Citations:

Zbl 0865.55008; Zbl 0555.55003
Full Text:

### References:

 [1] 10.2307/1970147 · Zbl 0096.17404 [2] 10.1093/qmath/12.1.52 · Zbl 0119.18701 [3] 10.1016/0040-9383(62)90097-6 · Zbl 0109.41402 [4] 10.1093/qmath/17.1.31 · Zbl 0136.43903 [5] 10.1093/qmath/17.1.165 · Zbl 0144.44901 [6] 10.1016/0040-9383(74)90012-3 · Zbl 0299.55008 [7] 10.1016/0040-9383(69)90003-2 · Zbl 0159.24801 [8] 10.1007/s11856-012-0085-1 · Zbl 1277.55003 [9] 10.4153/CJM-1979-107-2 · Zbl 0393.55012 [10] ; Harper, Localization in group theory and homotopy theory, and related topics. Lecture Notes in Math., 418, 58, (1974) [11] 10.5565/PUBLMAT_32188_05 · Zbl 0683.55004 [12] 10.2977/prims/1195175873 · Zbl 0645.55006 [13] ; Hemmi, Hiroshima Math. J., 24, 583, (1994) [14] ; Hemmi, Mem. Fac. Sci. Kochi Univ. Ser. A Math., 22, 59, (2001) [15] 10.2307/1995107 · Zbl 0181.51301 [16] ; Hubbuck, Illinois J. Math., 33, 162, (1989) [17] 10.2206/kyushumfs.38.285 · Zbl 0555.55003 [18] 10.2140/pjm.1980.87.373 · Zbl 0406.55018 [19] 10.1215/kjm/1250523329 · Zbl 0271.55009 [20] 10.2977/prims/1195189602 · Zbl 0383.22007 [21] 10.2307/1993608 · Zbl 0114.39402 [22] 10.2307/1993609 [23] ; Stasheff, Localization in group theory and homotopy theory, and related topics. Lecture Notes in Math., 418, 142, (1974) [24] 10.1007/BF01214031 · Zbl 0249.55017 [25] 10.2307/2039604 · Zbl 0245.55012 [26] 10.1007/BFb0070534 [27] 10.1016/0040-9383(70)90032-7 · Zbl 0191.53901 [28] 10.7146/math.scand.a-11075 · Zbl 0244.55011 [29] 10.7146/math.scand.a-11076 · Zbl 0244.55012 [30] 10.2307/1999447 · Zbl 0576.55009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.