I am doing a project for my Geometry class on a famous mathematician
named Jules Henri Poincaré.  To my understanding, he pioneered a
concept commonly known as Algebraic and/or Geometric Topology.  I have
come across various explanations for this concept, however I have had
trouble understanding them, due to my limited experiences with math.

Please explain this concept to me briefly but entirely in layman’s
terms so that my 10th grade geometry class can understand this
concept.

Hello! Thanks for using Google Answers

Congratulations on your choice of such a complex concept project at
the 10th grade mathematics level. It's always good to encounter a math
scholar.

From my mathematics bookmarks, I think I've found a resource that
offers the level of simplicity of explanation you're after. It's an
essay entitled "What is Topography", A Short and Idiosyncratic Answer,
by Robert Bruner.

A short excerpt demonstrates the understandability:

"In ordinary Euclidean geometry, you can move things around and flip
them over, but you can't stretch or bend them. This is called
"congruence" in geometry class. Two things are congruent if you can
lay one on top of the other in such a way that they exactly match.

In projective geometry, invented during the Renaissance to understand
perspective drawing, two things are considered the same if they are
different views of the same object. For example, look at a plate on a
table from directly above the table, and the plate looks round, like a
circle. But walk away a few feet and look at it, and it looks much
wider than long, like an ellipse, because of the angle you're at. The
ellipse and circle are projectively equivalent.

This is one reason it is hard to learn to draw. The eye and the mind
work projectively. They look at this elliptical plate on the table,
and think it's a circle, because they know what happens when you look
at things at an angle like that. To learn to draw, you have to learn
to draw an ellipse even though your mind is saying `circle', so you
can draw what you really see, instead of `what you know it is'.

In topology, any continuous change which can be continuously undone is
allowed. So a circle is the same as a triangle or a square, because
you just `pull on' parts of the circle to make corners and then
straighten the sides, to change a circle into a square. Then you just
`smooth it out' to turn it back into a circle. These two processes are
continuous in the sense that during each of them, nearby points at the
start are still nearby at the end."

http://www.math.wayne.edu/~rrb/topology.html

I hope this resource will be useful to you. 

`larre-ga

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Clarification of Answer by larre-ga on 19 May 2002 19:00 PDT

An additional simplified explanation of Algebraic Topology may be
found at:

http://www.math.rochester.edu:8080/u/jnei/algtop.html

I suggest that you turn your browser's image display to off, before
visiting this link. The images can take a long time to load. An
excerpt explains specifically algebraic topology using simple,
word-picture illustrations.  There's no direct author attribution,
though from the URL, the document originates from the Mathematics
Department of the University of Rodchester.

"To get an idea of what algebraic topology is about, think about the
fact that we live on the surface of a sphere but locally this is
difficult to distinguish from living on a flat plane. One way of
telling that we live on a sphere is to measure the sum of the three
angles of a triangle. For a small triangle, it is slightly more than
180 degrees. For a large triangle, it is much more. This tells us that
we live on a surface with what is called positive curvature. But,
since we can use small triangles, this is a local property, not a
global one. It properly belongs to the field known as differential
geometry. Algebraic topology is concerned with the whole surface and
points to the obvious fact that the surface of a sphere is a finite
area with no boundary and the flat plane does not have this property.
It expresses this fact by assigning invariant groups to these and
other spaces. Usually, these groups are something called homotopy
groups or another kind called homology groups. The groups are
invariant in the sense that they do not change if the space is
continuously deformed. The sphere is assigned an infinite group which
is a measure of the fact that the sphere has a hole in it and the
plane is assigned the zero group because it does not. The fact that
these groups are different tells us that the spaces are fundamentally
globally different. No doubt about it. Algebraic topology includes but
is not confined to the study of spaces of dimensions only two or
three. It includes, for example, the contemplation of the shape of the
three dimensional universe itself or even the contemplation of the
shape of the four dimensional space-time."

I hope this provides additional help for you.

The authors of both the reference works suggested have taken a
considerable amount of time and effort to present very complex ideas
at the layman's level. I have several additional resources, though
written for the college and post-graduate mathematics level, that I
would be happy to share with you if you want greater detail.

~L

Gave a good explanation of Topology, although could of applied it to
algebra a little more.  This will do, though.

Hi.
Another excellent (IMHO) introduction to the Subject of Geometry /
Algebraic Topology is at
http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Topology_in_mathematics.html
this site also has a biography of Poincare.
Probably if you want to tell someone what algebraic topology is about,
Euler's Königsberg bridge problem is excellent: Take a geometric
problem (finding a path in the city), derive mathematical (algebraic
with unknowns) formulae which a solution to the geometry problem would
fulfil, and proof that those don't (!) have a solution. Thus the
geometric problem does not have a solution. (There is no walk as asked
for in the K.b.p.
If you want to demonstrate why this is impossible, draw a graph: the
"islands" in K. as points and the bridges as edges. You find that 3
points ("knots") are connected to 3 edges (math-speak: "have degree
3") while the fourth knot has degree 5. However if you want to have a
path that uses all edges exactly once (called an "Euler path"), for
each knot you go through (after the frist and before the last), you
"walk" in on one edge and "walk" out on another. Thus the degree of
all points except the start and the endpoint must have degree
divisible by two because you leave as often as you arrive.
Thus a graph with 3 knots of degree 3 and one of degree 5 does not
have a Euler path. We have transformed the geometric question "is
there a path" to the algebraic question "what is the degree of the
knots".
(Be warned about the statement "A graph has a path traversing each
edge exactly once if exactly two vertices have odd degree." (on
MacTutor). It is wrong if you do not require of a path that the start
and end point are distinct.

Regards

Thomas

In response to the comment that the answer didn't mention anything
about applications to algebra, the meaning of the word algebra here
doesn't bear much relation to high school meaning of the word algebra.

Algebraic topology is about the study of algebraic objects that are
associated with topological spaces. Topological spaces and special
types of maps between them are the formal tools used by mathematicians
to study topology. An example of an algebraic object is a group (see
for instance http://members.tripod.com/~dogschool/groups.html).
Algebraic topology is about using objects such as groups to tell us
more about topological spaces.

You would need to wait until near the end of a university degree in
Mathematics or even until you are a postgraduate before you'd get a
chance to study algebraic topology, so it's not easy to describe in a
paragraph or two to a high school student!