• The Journal of Natural Sciences
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• v.10 no.1
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• pp.9-12
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• 1998

In this paper, we obtain that if $\psi : E \to F$ is a fibrewise fibration then postcomposition $\psi :C_B(Y, E) \to C_B(Y, F)$ is fibrewise fibration and if (X, A) is a closed fibrewise cofibration the the precomposition $\upsilon :C_B(X, E) \to C_B(A, E)$ is also a fibrewise fibration.

• Bulletin of the Korean Mathematical Society
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• v.36 no.4
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• pp.671-682
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• 1999

Using a base-point free version of the infinite symmetric product we define a fibrewise infinite symmetric product for any fibration $E\;\longrightarrow\;B$. The construction works for any commutative ring R with unit and is denoted by $R_f(E)\;l\ongrightarrow\;B$. For any pointed space B let $G_I(B)\;\longrightarrow\;B$ be the i-th Ganea fibration. Defining $M_R-cat(B):= inf{i\midR_f(G_i(B))\longrihghtarrow\;B$ admits a section} we obtain an approximation to the Lusternik-Schnirelmann category of B which satisfies .g.a product formula. In particular, if B is a 1-connected rational space of finite rational type, then $M_Q$-cat(B) coincides with the well-known (purely algebraically defined) M-category of B which in fact is equal to cat (B) by a result of K.Hess. All the constructions more generally apply to the Ganea category of maps.