Md Kutubuddin Sardar

University of Calcutta, Pure Mathematics, Department Member

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Papers by Md Kutubuddin Sardar

TOPOLOGY OF CONFIGURATION SPACE OF TWO PARTICLES ON A GRAPH, I

In this paper we study the homology and cohomology of configuration spaces F (Γ, 2) of two distin... more

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HOMOTOPY AND ISOTOPY PROPERTIES OF TOPOLOGICAL SPACES

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ON THE DEFORMATION OF AN n-CELL

By an extremely simple device, Professor Veblen (These PROCEIDINGS, 3, 1917, p. 655) has proved t... more By an extremely simple device, Professor Veblen (These PROCEIDINGS, 3, 1917, p. 655) has proved that every one-one continuous transformation of an n-cell and its boundary into themselves which leaves invariant all points of the boundary may be realized by a deformation. However, the deformation is of such a character that the boundary of the n-cell does not remain invariant but merely returns to its initial position as the deformation is completed. A slight modification of Professor Veblen's scheme gives a deformation during the entire course of which the boundary remains pointwise invariant. I shall give the proof only for the case n = 2, since the generalization to higher dimensions is immediate. Let the 2-cell and its boundary be represented by the interior and periphery of the unit circle C1. Any one-one continuous transformation T of the 2-cell into itself may then be expressed by a pair of equations in polar coordinates of the following form: * rl= R(r,w) pl (r, r,r) (T) where R and e are defined within the unit circle. Moreover, if the transformation is pointwise invariant on the boundary C1 of the 2-cell, we must also have R (1, q)=1, e (l,,r So) s. We shall extend the region of definition of the transformation. T over the entire plane by putting R (r, sp) = r, 0 (r, qP) = p, for (r _ 1). 406 PROC. N. A. S.

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ISOTOPY AND HOMEOMORPHISM

If X and Y are isotopically equivalent topological spaces, they are not necessarily homeomorphic.... more If X and Y are isotopically equivalent topological spaces, they are not necessarily homeomorphic. If X and Y are compact without boundary manifolds or «-pure simplicial complexes , then isotopy equivalence implies topological equivalence. An example of compact manifolds with boundaries which are not homeomorphic but are isotopically equivalent is given.

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