On the same n-types for the wedges of the Eilenberg-Maclane spaces

Abstract

For a connected CW-complex, we let SN T (X) be the set of all homotopy types [Y ] such that the Postnikov approximations X (n) and Y (n) of X and Y , respectively, are homotopy equivalent for all positive integers n. In 1992, McGibbon and Møller ([22, page 287]) raised the following question: Is SN T (ΣCP ∞ ) = * or not? In this article, we give an answer to the more generalized version of this query: The set of all the same n-types of the suspended wedge sum of the Eilenberg-MacLane spaces of various types of both even and odd integers is the set which consists of only one element as a single homotopy type of itself.2010 Mathematics Subject Classification. Primary 55P15; Secondary 55S37, 55P35, 55P40, 55P62, 16T05, 16W20.

On the Digital Cohomology Modules

“…These spaces were introduced in [10] and were used to characterize when a pointed CW-space has the homotopy type of a suspension ( [11] (Theorem A)); see also [12]. The author investigated the co-Hopf structures on a wedge of (localized) spheres [13][14][15][16][17][18] and a suspension with the standard comultiplication in the sense of same n-types [19][20][21][22][23]; see also [24,25] for the topics that are related to the fundamental concepts of those spaces.…”

Coalgebras on Digital Images

“…Co-Hopf spaces were introduced in [15] and were used to determine whether a pointed CW-space has the same homotopy type of the suspension of another pointed CW-space or not [16] [Theorem A]; see also [17]. The second author has developed the structures of a wedge of (localized) spheres as the co-Hopf spaces with various homotopy comultiplications [18][19][20][21][22][23], and the suspension structure with the standard comultiplication in the sense of same homotopy n-types [24][25][26][27][28]; see also [29,30] for the topics which are related to the fundamental concepts of those CW-spaces, and [31] for the equivariant homotopy theoretic point of view with the behavior of the local cohomology spectral sequence.…”

On the generalized same N-type conjecture

“…The answer to this question was also given in [14]: it is the one element set consisting of a single homotopy type. The answer to this question was also given in [14]: it is the one element set consisting of a single homotopy type.…”

“…2? The answer to this question was also given in [14]: it is the one element set consisting of a single homotopy type. More generally, what will happen in the case of the suspension of the smash products of the Eilenberg-MacLane spaces?…”