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At the present time, the average undergraduate mathematics major
finds mathematics heavily compartmentalized. After the calculus, he
takes a course in analysis and a course in algebra. Depending upon
his interests (or those of his department), he takes courses in
special topics. Ifhe is exposed to topology, it is usually
straightforward point set topology; if he is exposed to geom etry,
it is usually classical differential geometry. The exciting
revelations that there is some unity in mathematics, that fields
overlap, that techniques of one field have applications in another,
are denied the undergraduate. He must wait until he is well into
graduate work to see interconnections, presumably because earlier
he doesn't know enough. These notes are an attempt to break up this
compartmentalization, at least in topology-geometry. What the
student has learned in algebra and advanced calculus are used to
prove some fairly deep results relating geometry, topol ogy, and
group theory. (De Rham's theorem, the Gauss-Bonnet theorem for
surfaces, the functorial relation of fundamental group to covering
space, and surfaces of constant curvature as homogeneous spaces are
the most note worthy examples.) In the first two chapters the bare
essentials of elementary point set topology are set forth with some
hint ofthe subject's application to functional analysis."
In the past decade there has been a significant change in the
freshman/ sophomore mathematics curriculum as taught at many, if
not most, of our colleges. This has been brought about by the
introduction of linear algebra into the curriculum at the sophomore
level. The advantages of using linear algebra both in the teaching
of differential equations and in the teaching of multivariate
calculus are by now widely recognized. Several textbooks adopting
this point of view are now available and have been widely adopted.
Students completing the sophomore year now have a fair preliminary
under standing of spaces of many dimensions. It should be apparent
that courses on the junior level should draw upon and reinforce the
concepts and skills learned during the previous year.
Unfortunately, in differential geometry at least, this is usually
not the case. Textbooks directed to students at this level
generally restrict attention to 2-dimensional surfaces in 3-space
rather than to surfaces of arbitrary dimension. Although most of
the recent books do use linear algebra, it is only the algebra of
~3. The student's preliminary understanding of higher dimensions is
not cultivated.
In the past decade there has been a significant change in the
freshman/ sophomore mathematics curriculum as taught at many, if
not most, of our colleges. This has been brought about by the
introduction of linear algebra into the curriculum at the sophomore
level. The advantages of using linear algebra both in the teaching
of differential equations and in the teaching of multivariate
calculus are by now widely recognized. Several textbooks adopting
this point of view are now available and have been widely adopted.
Students completing the sophomore year now have a fair preliminary
under standing of spaces of many dimensions. It should be apparent
that courses on the junior level should draw upon and reinforce the
concepts and skills learned during the previous year.
Unfortunately, in differential geometry at least, this is usually
not the case. Textbooks directed to students at this level
generally restrict attention to 2-dimensional surfaces in 3-space
rather than to surfaces of arbitrary dimension. Although most of
the recent books do use linear algebra, it is only the algebra of
3. The student's preliminary understanding of higher dimensions is
not cultivated."
At the present time, the average undergraduate mathematics major
finds mathematics heavily compartmentalized. After the calculus, he
takes a course in analysis and a course in algebra. Depending upon
his interests (or those of his department), he takes courses in
special topics. Ifhe is exposed to topology, it is usually
straightforward point set topology; if he is exposed to geom etry,
it is usually classical differential geometry. The exciting
revelations that there is some unity in mathematics, that fields
overlap, that techniques of one field have applications in another,
are denied the undergraduate. He must wait until he is well into
graduate work to see interconnections, presumably because earlier
he doesn't know enough. These notes are an attempt to break up this
compartmentalization, at least in topology-geometry. What the
student has learned in algebra and advanced calculus are used to
prove some fairly deep results relating geometry, topol ogy, and
group theory. (De Rham's theorem, the Gauss-Bonnet theorem for
surfaces, the functorial relation of fundamental group to covering
space, and surfaces of constant curvature as homogeneous spaces are
the most note worthy examples.) In the first two chapters the bare
essentials of elementary point set topology are set forth with some
hint ofthe subject's application to functional analysis."
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