On Modular Cohomotopy Groups

DOI10.1007/S11856-022-2409-0arXiv2203.09105MaRDI QIDQ6393938

Publication date: 17 March 2022

Abstract: Let

p

be a prime and let

p i^{n} (X; m a t h b b Z / p^{r}) = [X, M_{n} (m a t h b b Z / p^{r})]

be the set of homotopy classes of based maps from CW-complexes

X

into the mod

p^{r}

Moore spaces

M_{n} (m a t h b b Z / p^{r})

of degree

n

, where

m a t h b b Z / p^{r}

denotes the integers mod

p^{r}

. In this paper we firstly determine the modular cohomotopy groups

p i^{n} (X; m a t h b b Z / p^{r})

up to extensions by classical methods of primary cohomology operations and give conditions for the splitness of the extensions. Secondly we utilize some unstable homotopy theory of Moore spaces to study the modular cohomotopy groups; especially, the group

p i^{3} (X; m a t h b b Z_{(2)})

with

d i m (X) l e q 6

is determined.

Mathematics Subject Classification ID

(H)-spaces and duals (55P45) Cohomotopy groups (55Q55)

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